Interesting analysis. This time I won’t start with the letter aleph.
Suppose a situation were described in which a philosopher proves to me intellectually why my faith is based on falsehood, refuting every theological claim I have in an intellectual/philosophical/logical way.
And for every answer that I / a rabbi gives him, he raises another 10 objections (let’s say it’s possible that there are no answers to all his difficulties). In that case, should I then "not believe"? "not keep the commandments"?

I see that there are believers in the religious community who, even if 1000 objections are raised against them and they have no answer, and indeed it seems that their faith is intellectually contradictory, will still continue to "believe"…
I think to myself that ideally, a person should go through life and base his faith on reason and logic.
It follows from that that if they prove to me that religion is not something exalted but merely a fiction and a nice story (and I would do very deep research for even several years if necessary) — then, in terms of truth, I should not believe…
What does the Rabbi think about this matter?

I don’t understand the question of whether you should believe. Ask yourself whether you believe or not. This is a decision a person has to make (see column 661). As for the meaning of proofs, see column 654. There I explained that it is not true that a proof is necessarily a conclusion one is compelled to accept.