Indeed, this proof seems to implicitly (and probably accidentally) assume a "Meinong's jungle" ontology. As Aristotle himself would agree, it is impossible for there to be a thing which does not exist (much like Kant, Aristotle argued that being is not a true category since it cannot be contrasted to a category of non-being).
The trouble with arguments that treat being or existence as properties that a thing can either have or not have is that they either conflate the concept of a thing with the thing itself (thus confusing the fact that the concept doesn't correspond to anything in the external world with the claim that the concept is something that doesn't exist) or they treat a sentence like, "Bigfoot does not exist" as saying "Bigfoot has the property of nonexistence" rather than "There is no such thing as Bigfoot". The confusion seems to result from ambiguities in the English language and the fact that we often use the same syntax to talk about existence as we do when talking about properties like "green", even though they must logically be treated in different ways.
Something tangentially related to this that might interest you is the problem of non-designating terms in logic. Classical logic assumes that all terms in the language refer to something that actually exists (so you cannot, for example, talk about Bigfoot while using classical logic). When using terms that may not refer to any actually existing thing, we have to use free logic. It seems to me that a lot of confusion coming from arguments regarding whether existence is part of something's essence could be cleared up if people understood the special logical character of existence and why you must use free logic for any proof that involves terms that are not presupposed to exist.

Yeah, taking in-depth mathematics definitely helps. As you would probably agree though, we just have to be careful in taking mathematics and applying it directly to reality insofar as mathematics is purely abstract in nature, and the material cosmos is not purely abstract.

The more I've reflected on this question the more I've come to conclude that an infinite amount of matter/energy seems possible with God as Creator, but this would not be possible part from God as classical theism understands Him. (Obviously, explaining why there is any physical cosmos at all ultimately leads to God, but trying to propose an infinite amount of physical things that came to existence from a finite amount of things without a Creator seems problematic.)

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But the second question that I'm still not quite sure about is if it is possible for a physical cosmos eternally existing in the past? Obviously God, creating ex nihilo, could have created an infinite amount of matter/energy sometime in the finite past. But could He have created an infinite amount of matter/energy, or even a finite amount of matter/energy, that always existed?

It seems we are left with 2 options:
(a) Either there is an infinite amount of changes reaching into the past.

(b) Or there was a completely changeless physical entity at some point that suddenly began to change.

I'm leaning towards the impossibility of both of these right now, but I'm always open to new thoughts on this!

Nice pun on infinity :) Again I'm not saying it's infinite with serious certainty, but I don't like seeing it ruled out in such a way. As another example, you can have an infinite universe with nothing but the same boring star over and over again (like an infinite series of 1's). I does help to have taken classes in advanced math (even though it's been a while), studying infinity mathematically was pretty interesting. Infinite series, for example, are quite useful, and used in engineering.

https://en.wikipedia.org/wiki/Series_(mathematics)

http://www.mathalino.com/reviewer/advance-engineering-mathematics/infinite-series

In many computing models, it's useful to represent parts of the model with infinite series, though the number in the series is limited by computing resources. It makes it easier to increase resolution with higher computing resources, however.

Well, I have to say, thanks to Michael and yourself, I think you may have intellectually converted me in regards to the possibility of an infinite amount of matter/energy existing!

So I am infinitely grateful to you for your patience and understanding. I know I am a slow thinker, so I give thanks to God for having gifted you with the needed patience to discuss this with me--and for your cooperation with that gift!

In the end, I guess Aquinas may be actually right in regards to the possibility of an eternally existing cosmos! I guess I should trust what Aquinas says a little more than I did (sorry Bonaventure for abandoning you)!

Well, I have to say, thanks to William and yourself, I think you may have intellectually converted me in regards to an infinite amount of matter/energy possibly existing!

So I am infinitely grateful to you for your patience and understanding. And I give thanks to God for having gifted you with that!

In the end, I guess Aquinas may be actually right in regards to the possibility of an eternally existing cosmos! I guess I should trust what Aquinas says a little more than I did!

Here is the only proposal that I am seeing as possibly coherent in regards to the hypothesis that an infinite amount of non-abstract entities exist:

Say God created, in a single moment, an infinite amount of matter/energy and it was all in a single form. This matter/energy began to act in accordance to its nature and it has taken varied forms at this point in history, including human persons.

For me, this is the only proposal that may possibly be coherent in regards to the proposal that the infinite exists in the non-abstract.

(The interesting thing is that it seems right now that you necessarily need God in regards to proposing that an infinite number of non-abstract entities exist.)

An infinite universe does not imply no new "things" (i.e. different forms of existing matter/energy) can't come into existence. Sure, you could exist somewhere else in the universe.

What do you mean specifically when you say "infinite universe"? What is actually infinite in an infinite universe that you are arguing for? We started this discussion by saying that there could be an infinite numbers of stars/planets.

We aren't that interested in the mathematical or abstract properties of infinite. Remember, we are talking about real physical, non-abstract, things right now. I completely agree with the abstract properties that you are proposing about infinity.

What needs to be argued for is that these abstract properties apply to non-abstract entities. This is what has led physicists, mathematicians, and philosophers to argue that complete/actual infinities do not exist in physical reality. (Of course there are some that also try to argue for the opposite, which is why we are having this discussion in the first place).

That is why this statement makes no sense when you try and apply it to physical objects, even though it can reasonably apply to abstract mathematical sets:

In other words, you can add things to an already infinite set and make it bigger.

This statement does not

Well obviously I don't believe in God but regardless of this I think you have already given the resolution here:

This would seem to suggest that all persons already exist. (Maybe we can get around this because of the fact that 'I', meaning my spiritual soul, is completely immaterial?)

Except I would phrase it as "the materials parts of all persons that will one day exist already exist" and fall back on spiritual souls being immaterial (if I believed in them)

I still haven't come across a good proposal as how to coherently hold that the unachievable (an infinite amount) can be achieved (an actual infinite set/amount).

What does "achieved" mean ?

Backtracking a little why do you assert:

(A) No new physical entities are coming into existence right now because an infinite amount of things already exists.

Based on your placement of "because" I assume you are somehow asserting that infinite = all ? It's quite possible to have an infinity of things but not all things. Imagine you have all the odd counting numbers 1, 3, 5, ... you don't have all the numbers because you are missing the even counting numbers 2, 4, 6, ...

Agree completely.