Are Theists Illogical for Believing in God?

[Purple Rabbit]

All math theorems have been proved by previous theorems and original axioms.

Axioms are by default unproven statements.
Anyway, 'proven' means proven in the language of a logical framework. This has no implicit bearing on universes.

(2+2=1 mod 3) does not exist in math, a correct modular statement is ( 2 + 2 ≡ 1 mod 3).
(≡) is the sign of congruence, not equality.

That's a matter of convention only.

Modular arithmetic is in fact applicable in our universe, the most common thing which involves modular math is the clock.

A clock is an abstraction of time by humans, not time itsel. Presenting it in a periodic cycle is a handy way to sync our practical daily use time to the rotation of the earth, but it is not applicable to time as a dimension of our universe. That is, as we currently understand it, a continuous timeline

All math theorems are proved by original axioms (ex. of axiom: 1+1=2).

1+1=2 is not an axiom of arithmetic. In fact, to prove 1+1=2 from axiom is a rather lengthy proof.

There is no such thing as "original" axioms. Axioms are unproven assumptions of mathematics. They don't necssarily relate to our reality. Famous example: euclidean geometry was based on five axiomas. Some were later shown not to hold in our universe. As a result Einstein and Minkovski replaced it with a new geometry.

Logic is very necessary for describing the universe, since all classical and modern physics involves logical math.

Yes. So what, I'm very familiar with physics. We do inded need it because we try to describe the world in terms of patterns and mathematics fits very good for that.

Well, a startling find with the discovery of quantum mechanics seems to suggest that this indeed is the case. At the quantum level of our very own universe even countability breaks down due to the Heisenberg Uncertainty Principle. In this realm a different logic, quantum logic, is used to describe the phenomena.

I don't see any reason of bringing quantum physics here, it only appears magical to public.
Same math axioms and theorems are used in quantum physics, you can see that in any physics book.
There is no 'our realm' and quantum realm, every macroscopic event can be explained by quantum physics, but the classical physics is used for that because it's simpler and gives a fair approximate answer.

Please don't parrot some lame argument about the magical connotation of QM when it's not appropiate. I've studied quantum physics. This is not an argument that suggests some mystical phenomenon from QM. All I say is that QM can be described with a logic that's different from the normal stuff. That is quantum logic. Just google it. It exists and has practical scientific use in constructing quantum computers. I assume you know that quantum physics currently hasn't been unified with general relativity? If not google it. The underlying math is different. And that is a real enigma. If you can unify these realms, you can make the headlines tomorrow. This is the holy grail of physics.

What this shows is that you cannot conclude from logic that it has a counterpart in reality and vice versa that you can't conclude that any possible reality necessarily is a reality that obeys one unified logical framework, let alone the logical framework we might arrive at for our universe.

Logic is not a rule or something to be obeyed. Logic is not something that have properties, because you say quantum logic, different logics...
Logic is simply the study of reasoning. In all universes, you can reason and use your intellect and therefore think and deduce logically.

I see no argument there. You're just asuming a whole lot on logic in other universes.
I suggest you read up on Euclidean Geometry and it's replacement by general relativity. Should be very enlightening.