"Correct me if I am wrong, but it is the duty of any Catholic who has the slightest doubt that, say, the Church is right about Jesus to do everything in his or her power to convince himself or herself that the Church is indeed right about Jesus".

At this point it is already over.

For atheists, this is a chess game, where frequently the win of a single pawn can mean inevitable victory or resignation. For Catholics, it is more akin to any sport which uses force to overcome the endurance of the opponent.

Thanks. The Catechism is always the first place to look, and the quotes are helpful. But I was also hoping to have some discussion of what assent means in practice. We have three concepts under discussion, assent, doubt, and denial. As I said above, there is a saying that a thousand difficulties do not add up to a doubt. Clearly, there are many dogmas (for example, the Trinity) that no one really understands, so it seems clear to me that a faithful Catholic can say, "I can't even begin to conceive of how there can be three Persons in one God," but nevertheless assent to it. Lack of understanding may be a difficulty, but it is not a doubt. But what about Adam and Eve and Original Sin? Original Sin is a dogma, but are Catholics required to assent to the story of Adam and Eve?

I have an old story from high school which I won't repeat here, but the punchline was that, after one of my teachers (a Christian Brother) was challenged in an ongoing debate that lasted over a period of time, the teacher finally ended the debate by saying, "I can't explain it, but this is what you have to believe." What if there is something you don't believe and you are told you have to?

"I'm just one hundred and one, five months and a day."

"I can't believe that!" said Alice.

"Can't you?" the Queen said in a pitying tone. "Try again: draw a long breath, and shut your eyes."

Alice laughed. "There's no use trying," she said: "one can't believe impossible things."

"I daresay you haven't had much practice," said the Queen. "When I was your age, I always did it for half-an-hour a day. Why, sometimes I've believed as many as six impossible things before breakfast."

Perhaps the concept that needs more explanation is  religious assent .

There are two "bodies" of teachings by the Magisterium (both of which are to be adhered to as Catholics):
1). "Divinely Revealed", and as teachings of Christ, (and thus infallible). It is the content of the faith itself.
2). "Ordinary" teachings (in matters of faith and morals)
of the Magisterium derived from Divine Revelation.

Divine revelation is assented to through faith, ordinary teachings through religious assent.
(Religious assent is an extension of the assent of faith).

From the catechism (my bold):

891 "The Roman Pontiff, head of the college of bishops, enjoys this infallibility in virtue of his office, when, as supreme pastor and teacher of all the faithful - who confirms his brethren in the faith he proclaims by a definitive act a doctrine pertaining to faith or morals. . . . The infallibility promised to the Church is also present in the body of bishops when, together with Peter's successor, they exercise the supreme Magisterium," above all in an Ecumenical Council. When the Church through its supreme Magisterium proposes a doctrine "for belief as being divinely revealed," and as the teaching of Christ, the definitions "must be adhered to with the obedience of faith."  This infallibility extends as far as the deposit of divine Revelation itself.

892 Divine assistance is also given to the successors of the apostles, teaching in communion with the successor of Peter, and, in a particular way, to the bishop of Rome, pastor of the whole Church, when, without arriving at an infallible definition and without pronouncing in a "definitive manner," they propose in the exercise of the ordinary Magisterium a teaching that leads to better understanding of Revelation in matters of faith and morals. To this ordinary teaching the faithful "are to adhere to it with religious assent" which, though distinct from the assent of faith, is nonetheless an extension of it.

The 'mystery' is the concept that needs meditation. If one cannot accept mystery, can one accept anything new under the sun?

There are many shades of doubt as there are of denial and assent.

I think denial is a reasonably clear concept. Doubt may be more complex, but I quoted the Catechism using the words "obstinate doubt," and I don't think that is particularly difficult to conceive of.

Denial of dogmatic mystery at its most extreme, rejection without reason, and continual (repeated over time, often public, glorying in one's rebellion--such) obstinacy would be heresy.

I think you are defining heresy much too narrowly. For a Catholic, denial of dogma is heresy. Rejection with reason is heretical, not merely rejection without reason. Perhaps the concept that needs more explanation is religious assent.

There are many shades of doubt as there are of denial and assent. There are many mysteries involved in how our lives intermingle with the life of our Lord and Savior and Creator. Asking the Lord for understanding of some mystery is the preferred method of resolving questions or failure to understand. There is plenty of evidence in Scripture for many mysteries, much of it esoteric but to those who have ears to hear and to those who wish their eyes to be opened. Like most everything else in life, we don't understand unless we study, meditate, and remain docile to the teaching of experts, to the Creator of all experts, and to the satisfaction of our own reason. Denial of dogmatic mystery at its most extreme, rejection without reason, and continual (repeated over time, often public, glorying in one's rebellion--such) obstinacy would be heresy. One is free to commit it but iIn doing so, one has denied scripture and the 2000 year teaching of Jesus who granted to Peter the keys to His kingdom.

You are not correct in ascribing sin to that person.

I do not ascribe sin to someone who finds no evidence in the Bible for the Bodily Assumption of Mary. There is no evidence in the Bible! I would ascribe sin to someone who said, "Since there is no evidence in the Bible, I do not believe in the dogma of the Assumption." I ascribe sin (actually, heresy) to any baptized Catholic who denies a dogma of the Catholic faith. That is the very definition of heresy, as quoted from the Catechism above:

Heresy is the obstinate post-baptismal denial of some truth which must be believed with divine and catholic faith, or it is likewise an obstinate doubt concerning the same . . . .

OK. Let's assume that a Catholic finds no evidence in the Bible for the bodily assumption of the Blessed Virgin into heaven. You are not correct in ascribing sin to that person.

We may inquire if dogma is true because there is nothing to stop the questioning.

There is conscience. Dogma requires assent of faith. As it says in my chart above:

Assent of Faith—The believer makes an act of faith, trusting that this teaching is revealed by God.

If you want to remain a faithful Catholic, any inquiry into dogma must begin with assent of faith. The inquiry must be not whether a dogma is true, but how it is true and why the Church teaches it. The truth of dogma may be examined, but it may not be questioned or denied. That is what makes it dogma.

I think you are basically wrong—and three years late! ;-)

Catholics are certainly free to study anything they want about the Catholic faith to any degree they want, but they are not permitted to say, "I can't accept this dogma. I'm going to shop around for something I can accept." They certainly can say, "I find this dogma difficult to accept. I am going to keep on studying it until I can find some way to accept it." From the Catechism:

Faith

2087 Our moral life has its source in faith in God who reveals his love to us. St. Paul speaks of the "obedience of faith"9 as our first obligation. He shows that "ignorance of God" is the principle and explanation of all moral deviations.10 Our duty toward God is to believe in him and to bear witness to him.

2088 The first commandment requires us to nourish and protect our faith with prudence and vigilance, and to reject everything that is opposed to it. There are various ways of sinning against faith:

Voluntary doubt about the faith disregards or refuses to hold as true what God has revealed and the Church proposes for belief. Involuntary doubt refers to hesitation in believing, difficulty in overcoming objections connected with the faith, or also anxiety aroused by its obscurity. If deliberately cultivated[,] doubt can lead to spiritual blindness.

2089 Incredulity is the neglect of revealed truth or the willful refusal to assent to it. "Heresy is the obstinate post-baptismal denial of some truth which must be believed with divine and catholic faith, or it is likewise an obstinate doubt concerning the same; apostasy is the total repudiation of the Christian faith; schism is the refusal of submission to the Roman Pontiff or of communion with the members of the Church subject to him."

To take an example, if a Catholic should take the position that there is no evidence in the Bible—in fact, no evidence at all—for the bodily assumption of the Blessed Virgin into heaven, and he or she even goes so far as to say, "I don't believe it, but it makes no practical difference to anything in my life as a Catholic," that is sinful. On the other hand, if he or she says, "I don't see how it can be true, and it is very difficult to accept, but it is not up to me to question the Church, so I will just not dwell on it," that is religious assent.

Of course, there are no Catholic thought police tasked with discovering what is going on inside the heads of Catholics. In that sense, a Catholic is free to think anything he or she wants. It is up to the individual Catholic to "police" his or her own thoughts and to take some kind of action if difficulties become doubts. (There is an old saying that a thousand difficulties don't add up to a single doubt.)