Other religions don't have Jewish law?

Islam does, but of course it's different, and of course mistaken.

Aside from the fact that Judaism is the true one, why is it more correct and better, if at all? (In this world.)

Aside from the fact that it is true, why is it correct? I don't understand.
What does "better" mean? I don't understand that either.

Is Judaism the recipe for a happy life even in this world?

I don't think so.

Is character refinement a supreme value in Judaism more than in other religions?

I don't know other religions, so I can't answer. In Judaism itself, the status of character refinement is not clear. Rabbi Chaim Vital asked why there is no commandment about it (and he ignores the commandment to cleave to the attributes of the Holy One, blessed be He). I discussed this here on the site, but it is not relevant to these comparative questions.
I don't know what these questions are driving at. Are you engaged in the comparative study of religions? I have no interest whatsoever in such questions.

In Orchot Tzaddikim it is written that the entire Torah depends on character refinement. According to that, it comes out that this is the highest value in Judaism. Does the Rabbi agree?

No, but I won't answer here any further until you explain to me where this discussion is going.

Every person has a psychological need for spirituality, and religion provides it through prayer, study, and fulfilling commandments. There is a person who is religious in order to satisfy this need, and there is a person who is religious not only for that need but also in order to positively change his traits and character. I thought that perhaps this is the essential difference between Judaism and the other religions, since it is impossible to fulfill God's will according to our Torah without working on one's character and changing oneself. Does the Rabbi agree, and if not, why?

A person who is religious in order to attain something, spiritual or material, is not serving God. A servant of God is supposed to serve Him because that is what is true and proper—not in order to improve his traits, and not in order to satisfy experiential needs or any other needs.