atheists will make an exception, postulating nothing as a cause.

Talk about a massive strawman! No atheist has ever done this. Not a single one. If someone tells you, "Nothing caused the Universe," they are absolutely not claiming, "There is a cause of the Universe, and that cause is nothing." They are claiming, "There is no cause of the Universe." Your conflation of these two completely different views is, well, in your own words, enough to "make a first-year philosophy student cringe." After all, even first-year philosophy students should already know that to make a good argument, they have to represent opposing views accurately, and I know that you know this as well - so you should no better than to write such a bizarre contortion of what some atheists actually believe.

Atheists know that anything which begins to exist must have a cause of its existence.

I'm sorry, but you cannot win an argument by asserting that your opponent already knows you are right. That tactic prevents any honest debate from occurring.

Science is founded on the principle that “things cause things.”

This is simply false. The principle that "things cause things" has nothing to do with the scientific method at all. You can't just make up principles and claim that they are part of the foundation of science - again, that's not how honest debate works. Furthermore, science has actually thrown the maxim that everything that begins to exist has a cause into question. Virtual particles begin to exist without a cause. So do electron-positron pairs that high-energy photons can spontaneously turn into. In beta decay, a quark emits a W boson without any cause, and the boson decays into an electron and an antineutrino without any cause. So clearly things do begin to exist without a cause. And remember, what is at issue here is not whether these things can come into existence out of nothing, but whether they can come out of existence without a cause. You already believe that something can come into existence out of nothing because that's exactly what you believe about the Universe, that it was created out of nothing. What's at issue is whether the Universe needs a divine cause for its coming into existence, or whether it could have come into existence uncaused, and the examples from particle physics clearly show that things can come into existence uncaused. Pointing out that they came from preexisting matter is a completely different issue.

The reason God's complexity poses a problem is not that the Universe can't have been caused by something complex. The problem is that you can't explain the complexity of the Universe by pointing to something even more complex. Not only does that get you nowhere - it actually sets you back: Now you are left with an even bigger complexity problem to solve.

As for your seven objections to God's complexity:
1. Just because Christians claim their God is complex doesn't mean that the God they describe actually is. If I told you that the Universe is actually a simulation in a giant supercomputer, you likely reject this on the basis of Occam's razor. But what if I claimed that it was a perfectly simple supercomputer? Would you accept my explanation as the simplest possible explanation of reality? Of course not! You would scoff at the idea because you know that a supercomputer capable of simulating all of reality can't be simple. Similarly, a mind that knows all truths about reality, is capable of acting and making decisions, can do anything logically possible, and is thinking about every single particle in existence at every moment in time, would have to be infinitely complex. And far from anthropomorphizing God, atheists who point this out recognize the massive gulf between man and God. Finite human consciousness is already extremely complex, so an infinite consciousness would be infinitely more complex. You can claim that God is simple all you want, but if the God you describe is not actually simple, claiming he is reduces your worldview to incoherence.
2. How does being pure spirit or devoid of parts make something perfectly simple? Furthermore, if it does make something perfectly simple, then all this means is that a conscious mind can't be pure spirit or devoid of parts. A conscious mind can't be perfectly simple because the nature of consciousness is extremely complex. It is therefore true by definition that a perfectly simple entity can't be conscious, let alone an infinite consciousness capable of imagining, doing, and thinking infinitely many things. Dr. Craig's objection to this argument also doesn't work. The ideas that a mind has are mental states. A mind cannot possibly be simpler than the ideas it has because the ideas are part of the mind. Having ideas is a conscious activity, not just a result of conscious activity.
3. Being understandable and being simple are not the same thing. We all understand what a human is quite easily, but does that make humans simple? We can even comprehend the concept of an infinitely complex computer doing infinitely many calculations at once. You are confusing the concept of a thing with the thing itself and ease of understanding with simplicity.
4. The "simple to complex" rule is not a real rule at all. As I stated at the beginning of this comment, the problem with God's complexity isn't that it makes it impossible to create the Universe. It's that it makes God useless as an explanation. Moreover, if you accept Occam's razor, it makes God a priori unlikely. If you think atheists who point out God's complexity are doing it because of a "simple to complex" rule, you don't understand their objections.
5. More of the same problem. Many theists argue that it is far too unlikely for something as complex as the Universe to exist by chance or as brute fact. But, if this is true, then it is even more unlikely for God to just exist without an explanation, since he is even more complex. The example of a book and its author isn't a counterexample to this objection. A human author would be a terrible explanation for why a book exists if the human author's existence was a brute fact - then you would be explaining something that is unlikely to be a brute fact (the existence of a complex book) in terms of something even less likely to be a brute fact (the existence of an even more complex human). But the existence of humans can be explained by the existence of much simpler things. But since theists assert that God's existence isn't explained by any other facts, we can't do the same thing for God, making him a very unreasonable explanation for anything. The probability of God just existing is like the probability of a Boltzmann brain coming into existence for no reason - except, it is actually much lower than that because a Boltzmann brain is only a finite mind, whereas God is an infinite one.
6. "An effect cannot be greater than its cause," is a meaningless statement. Peter Kreeft's assertion that evolution is an example of design is just that - an assertion - and nothing more.

The evolutionary process seems to know where it's going.

Try saying this to any evolutionary biologist, and you’ll be laughed out of the room. Your statement is the exact opposite of what evolutionary theory says. Evolution does not have an end goal in mind - the idea that it does is pure science fiction. Here is what actual biology professors have to say about it: "Most importantly, evolution does not progress toward an ultimate or proximate goal. Evolution is not 'going somewhere'; it just describes changes in inherited traits over time." https://www.nature.com/scitable/knowledge/library/evolution-is-change-in-the-inherited-traits-15164254/

If such a 'simple to complex' process exists then what set it in motion?

This is the God of the gaps fallacy. Just because we don't currently know how life began doesn't mean God did it.

What mechanism keeps it on course?

We've known for over a century and a half now that natural selection is the mechanism for evolution. Natural selection is an unguided process, so asking, "What guides natural selection?" is incoherent. And no, that doesn't make it look like a "transcendent, intentional and intelligent cause". Not only is that a completely unjustified assertion, it's the exact opposite of the truth - an unguided process is, by definition, not intentional or intelligent. And natural selection is not transcendent either - you seem to be treating "natural selection" and "evolution" as if they mean the same thing and are both black boxes that you can insert whatever explanation you want into. This is a complete misunderstanding of the basics of how evolution works. Evolution is not a black box. We know exactly how it happens. If you want to understand how it happens, you'll need to take a biology class, not a theology class. Natural selection refers to the main mechanism by which evolution occurs. It doesn't make sense to ask "How does natural selection occur?" because natural selection is the "how". Natural selection is a process, not the result of one. And again, natural selection is not some mysterious force - it's a precise mechanism. So you can't say that natural selection is actually just some thinly veiled description of God.

Final thoughts
God's name being simple doesn't make God simple.
Science must proceed for its own sake, not for the sake of the Christian apologetic. That's the only way science can proceed. If you are only doing science to try to prove a particular predetermined conclusion, then you aren't doing science at all. A perfect example of that was your sixth point above - rather than giving an accurate explanation of evolution and natural selection, you, in attempting to stretch them into something that confirms your worldview, completely butchered concepts that are not even hard to understand. I know that you are capable of understanding evolution, but you clearly didn't understand what you were talking about when writing that part of the article, and that was because, rather than actually trying to learn the science, you were more focused on using it for the sake of apologetics.
If your God is the God that "provides an explanation for those things that cannot be explained by science," then he is the god of the gaps. That’s literally just the definition of “god of the gaps”.
"Nothing" is not the explanation for everything. Putting aside the question of how on Earth you can conclude that, if there is an explanation for everything, it must be God, the proposition "There is no explanation for everything," means something completely different from, "'Nothing' is the explanation for everything." In fact, the former implies the falsehood of the latter. I hold that there is no explanation for everything. In fact, an explanation for everything is logically impossible, and this is easy to see: If everything can be explained by something necessary, then everything is necessary (because a full explanation must be something that entails the explanandum). However, not everything is necessary, so, if there is an explanation for everything, it must be contingent. However, if the explanation for everything is contingent, it is not truly an explanation for everything after all because it can't explain itself. QED.

As for your seven objections to God's complexity:

1. Just because Christians claim their God is complex doesn't mean that the God they describe actually is. If I told you that the Universe is actually a simulation in a giant supercomputer, you likely reject this on the basis of Occam's razor. But what if I claimed that it was a perfectly simple supercomputer? Would you accept my explanation as the simplest possible explanation of reality? Of course not! You would scoff at the idea because you know that a supercomputer capable of simulating all of reality can't be simple. Similarly, a mind that knows all truths about reality, is capable of acting and making decisions, can do anything logically possible, and is thinking about every single particle in existence at every moment in time, would have to be infinitely complex. And far from anthropomorphizing God, atheists who point this out recognize the massive gulf between man and God. Finite human consciousness is already extremely complex, so an infinite consciousness would be infinitely more complex. You can claim that God is simple all you want, but if the God you describe is not actually simple, claiming he is reduces your worldview to incoherence. 

2. How does being pure spirit or devoid of parts make something perfectly simple? Furthermore, if it does make something perfectly simple, then all this means is that a conscious mind can't be pure spirit or devoid of parts. A conscious mind can't be perfectly simple because the nature of consciousness is extremely complex. It is therefore true by definition that a perfectly simple entity can't be conscious, let alone an infinite consciousness capable of imagining, doing, and thinking infinitely many things. Dr. Craig's objection to this argument also doesn't work. The ideas that a mind has are mental states. A mind cannot possibly be simpler than the ideas it has because the ideas are part of the mind. Having ideas is a conscious activity, not just a result of conscious activity. 

3. Being understandable and being simple are not the same thing. We all understand what a human is quite easily, but does that make humans simple? We can even comprehend the concept of an infinitely complex computer doing infinitely many calculations at once. You are confusing the concept of a thing with the thing itself and ease of understanding with simplicity.

4. The "simple to complex" rule is not a real rule at all. As I stated at the beginning of this comment, the problem with God's complexity isn't that it makes it impossible to create the Universe. It's that it makes God useless as an explanation. Moreover, if you accept Occam's razor, it makes God a priori unlikely. If you think atheists who point out God's complexity are doing it because of a "simple to complex" rule, you don't understand their objections.

5. More of the same problem. Many theists argue that it is far too unlikely for something as complex as the Universe to exist by chance or as brute fact. But, if this is true, then it is even more unlikely for God to just exist without an explanation, since he is even more complex. The example of a book and its author isn't a counterexample to this objection. A human author would be a terrible explanation for why a book exists if the human author's existence was a brute fact - then you would be explaining something that is unlikely to be a brute fact (the existence of a complex book) in terms of something even less likely to be a brute fact (the existence of an even more complex human). But the existence of humans can be explained by the existence of much simpler things. But since theists assert that God's existence isn't explained by any other facts, we can't do the same thing for God, making him a very unreasonable explanation for anything. The probability of God just existing is like the probability of a Boltzmann brain coming into existence for no reason - except, it is actually much lower than that because a Boltzmann brain is only a finite mind, whereas God is an infinite one. 

6. "An effect cannot be greater than its cause," is a meaningless statement. Peter Kreeft's assertion that evolution is an example of design is just that - an assertion - and nothing more. "The evolutionary process seems to know where it's going." I dare you to say this to any evolutionary biologist. Your statement is the exact opposite of what evolutionary theory says. Evolution does not have a goal in mind - the idea that it does is pure science fiction. Here is what actual biology professors have to say about it: "Most importantly, evolution does not progress toward an ultimate or proximate goal. Evolution is not 'going somewhere'; it just describes changes in inherited traits over time."
https://www.nature.com/scitable/knowledge/library/evolution-is-change-in-the-inherited-traits-15164254/
"If such a 'simple to complex' process exists then what set it in motion?" This is the God of the gaps fallacy. Just because we don't currently know how life began doesn't mean God did it. "What mechanism keeps it on course?" We've known for over a century and a half now that natural selection is the mechanism for evolution. Natural selection is an unguided process, so asking, "What guides natural selection?" is incoherent. And no, that doesn't make it look like a "transcendent, intentional and intelligent cause". Not only is that a completely unjustified assertion, it's the exact opposite of the truth - an unguided process is, by definition, not intentional or intelligent. And natural selection is not transcendent either - you seem to be treating "natural selection" and "evolution" as if they mean the same thing and are both black boxes that you can insert whatever explanation you want into. This is a complete misunderstanding of the basics of how evolution works. Evolution is not a black box. We know exactly how it happens, and you will learn how it happens in any biology class. Natural selection refers to the main mechanism by which evolution occurs. It doesn't make sense to ask "How does natural selection occur?" because natural selection is the "how". Natural selection is a process, not the result of one. And again, natural selection is not some mysterious force - it's a precise mechanism. So you can't say that natural selection is actually just some thinly veiled description of God.

Final thoughts

God's name being simple doesn't make God simple.

Science must proceed for its own sake, not for the sake of the Christian apologetic. That's the only way science can proceed. If you are only doing science to try to prove a particular predetermined conclusion, then you aren't doing science at all. A perfect example of that was your sixth point above - rather than giving an accurate explanation of evolution and natural selection, you, in attempting to stretch them into something that confirms your worldview, completely butchered concepts that are not even hard to understand. I know that you are capable of understanding evolution, but you clearly didn't understand what you were talking about when writing that part of the article, and that was because, rather than actually trying to learn the science, you were more focused on using it for the sake of apologetics. 

If your God is the God that "provides an explanation for those things that cannot be explained by science," then he is the god of the gaps.

"Nothing" is not the explanation for everything. Putting aside the question of how on Earth you can conclude that, if there is an explanation for everything, it must be God, the proposition "There is no explanation for everything," means something completely different from, "'Nothing' is the explanation for everything." In fact, the former implies the falsehood of the latter. I hold that there is no explanation for everything. In fact, an explanation for everything is logically impossible, and this is easy to see: If everything can be explained by something necessary, then everything is necessary (because a full explanation must be something that entails the explanandum). However, not everything is necessary, so, if there is an explanation for everything, it must be contingent. However, if the explanation for everything is contingent, it is not truly an explanation for everything after all because it can't explain itself. QED.

You also claim, "Atheists know that anything which begins to exist must have a cause of its existence." I'm sorry, but you cannot win an argument by asserting that your opponent already knows you are right. That tactic prevents any honest debate from occurring. 

You claim, "Science is founded on the principle that 'things cause things.'" This is simply false. The principle that "things cause things" has nothing to do with the scientific method at all. You can't just make up principles and claim that they are part of the foundation of science - again, that's not how honest debate works. Furthermore, science has actually thrown the maxim that everything that begins to exist has a cause into question. Virtual particles begin to exist without a cause. So do electron-positron pairs that high-energy photons can spontaneously turn into. In beta decay, a quark emits a W boson without any cause, and the boson decays into an electron and an antineutrino without any cause. So clearly things do begin to exist without a cause. And remember, what is at issue here is not whether these things can come into existence out of nothing, but whether they can come into existence without a cause - you already believe that something can come into existence out of nothing because that's exactly what you believe about the Universe, that it was created out of nothing. Therefore, pointing out that these particles came from preexisting matter and energy doesn't refute this objection - that just means they didn't come from nothing, not that they were caused.

Yes, I've read that as well, and I think that's what Krauss says as well, but other physicists have correctly pointed out that, however close to "nothing" a quantum vacuum is, it's still something, and it still presupposes the laws of (quantum) physics. Which only postpones the question one stage.

I've no idea. I didn't write the book. Neither have I read it. But from what I've read about it if I had to hazard a guess I would say that the quantum vacuum from a physical point of view is remarkably close to what most people would call physical nothing.

Yes, Krauss has his own private interpretation of nothing. http://edwardfeser.blogspot.com/2013/02/forgetting-nothing-learning-nothing.html

Which comes from? And why call it "nothing"?

It is because I reject Platonism that I also reject the transcendent.

True, but some folks always want to shift things to how the universe functions rather than that the universe is. If you are trying to explain lightning, you can happily cite various electrical properties of air and ground. But if you are trying to explain why those properties exist and possess those relations, you are looking for a different level of explanation. That is why "Goddidit!" is an inadequate explanation for secondary causation. God is supposed to be primary causation.

The universe exists iff at least one of the things that comprise it exist. The sum itself does not require a reason. What is the reason for the Douggeorge. the set {Doug Shaver, George}? Doug has a reason for existing, George has a reason for existing, but the congruence of Doug+George need not have a reason. That is, it is the existence that requires a reason.

Given that nothing can give existence to itself -- if it could, it would already have existence, since it cannot give what it does not have -- the ultimate reason for existence must be transcendent. It cannot be "the laws of physics," because there cannot be laws of "physics" without something "physical" to describe. (In the Modern era, laws of nature are phenomenological, not causal, and are generally mathematical. And unless one is a Platonist, numbers as such have no causative power. If one is a Platonist, then we still wind up with something transcendent.