It doesn't matter. The point is, if you had solid evidence at hand for the existence of a particular god (regardless of whether it is concrete or abstract), would you use it?
This question has already been answered.
Your answer:
Okay, good. Now, if there were evidence to support your proof of said god, would you use it?
Depends on the context. If the context demands evidence, then of course.
Would this context demand evidence?
What is the context? When you ask for proof of existence, what effect is this evidence to inform? Materialist? Idealist? Or does the distinction not matter at all?
I will explain this to you again. You (as a theist) are trying to prove the existence of your particular god. You find strong, solid evidence (physical artifact, logical proof, etc.) for the existence of your particular god. Would you use said evidence?
Once again this has already been answered. In addition to that which I've emboldened, I also stated this:
If I were to coach one on presenting an argument for the existence of God, I would inform them to highlight the significance abstracts have in rationalizing existence, and that it's logically dissonant to claim that abstracts, whether "rigorously defined" or "purely imaginative," do not "exist," while using abstracts as a gauge for determining that which exist and that which does not. I would also coach them to point out that since we possess and are possessed by our own minds, we'd have to be able to control for an existence independent of our mind's bias--meaning, you isolate it, and you observe it's behavior. How do you perceive without the influence of your mind?
If you cannot answer that question in a straightforward way, then let me ask you this:
Nothing roundabout in my responses.
Is it true that the more evidence/proof one has to support their premise, the better (more convincing) their premise will be?
No.
If an argument contains a logical fallacy (say, circular reasoning), then is it still valid (from a general POV)?
It would be logically inconsistent; validity is another matter.
Since you stated that not everyone bases validity on logical consistency, then it follows that not everyone would believe that logical fallacies would invalidate an argument. Is this what you are saying?
No. To engage argument is to tacitly or explicitly accept logic. Not all Theistic statements are arguments; case in point: I believe God exists. It's not an argument at all; it's a statement of truth. Now, let's add a reason. I believe God exists because I feel God in my soul. Once again, another true statement. How do you scrutinize that on the basis of logical consistency? Remember logic doesn't create truth; it connects true statements inductively, or deductively.
So you are saying that mathematics, science, and logic are completely made up. Correct?
I said they were imaginary. Then again, they are no less conceived by the mind than that to which you refer to as "made-up."