What about large parts of Kabbalah, especially the later Kabbalah? Doesn't the Rabbi see something strange there?
A book that came out of nowhere, wasn't mentioned anywhere, and is suddenly attributed to Rashbi; the wife and daughter of the man who published it even claimed that he made it up out of his own head for money; all researchers are unanimous that it is later; and even sages thought that large parts of it were copied, as the Ya'avetz showed in Mitpachat Sefarim.

Beliefs that were clearly influenced by Greek philosophies, mainly Platonic and Neoplatonic ones, and also some similarity to the beliefs of Gnostic sects (even if it's unclear how much they were influenced by those sects, if at all).
Ideas that were clearly taken from Muslim alchemists; beliefs such as reincarnation of souls, which have no trace in the Sages and which Geonim and medieval authorities opposed; and absurdly, the earliest believers in reincarnation were Second Temple sects, while the first documented source after their disappearance was Anan ben David, founder of the Karaite sect, or one of its founders.
Likewise, there are contradictions between Rashbi in the Mishnah and the "Rashbi" of the Zohar.

Does Kabbalah really have an ancient tradition? Aren't there problematic beliefs here? After all, there are many anthropomorphic descriptions there. Many do not take them literally, but quite a few kabbalists did believe them literally. There are also Jewish laws there that contradict the Talmud and the medieval authorities, and were ruled in opposition to all accepted rules of Jewish law based on books that "fell" from nowhere. Even a prophet cannot introduce new laws for future generations.
Is it really serious to rely on such problematic traditions and chains of transmission?

You are assuming quite a few factual assumptions regarding the Zohar, and although I haven't checked the matter thoroughly, in my opinion they are disputed. In any case, those who give it validity believe that the tradition about it is reliable. The contents can always be worked out.
Correlations to the views of other groups do not mean much. First, because it is legitimate to be influenced by and draw from others (that certainly happened in the revealed part of Torah among the Sages as well, and in every generation), and also because sometimes the direction of influence could have been the reverse (from the Torah to others).
I think the discussion about other people's views is unnecessary and unhelpful. If you are interested in someone's opinion, ask him.
I can only tell you my own opinion, if there is a concrete question here.

Regarding the Zohar, there is not a single researcher who thinks it is from the time of Rashbi. There are some disputes about the identity of its author or authors, but everyone agrees that it is later than Rashbi.
It is also possible to see contradictions between Rashbi's view in the Mishnah and the view presented in the Zohar, a basic lack of understanding of the geography of the Land of Israel, linguistic corruptions, and more.
And there is also documentation that the wife and daughter of de Leon claimed that he invented it out of his own heart.

Influences can go in the opposite direction, but in my opinion it is much more likely that the kabbalists were influenced by Greek philosophy. Especially since the earliest documentation of these ideas was among Hellenists and adherents of the Greek-Roman mystery religions, and the Greek philosophers on whom they were based apparently lived without significant contact with the Jews, while the connection of the ancient Near East to their ideas came after the conquests of Alexander the Great.
One could also argue that Little Red Riding Hood lived 200,000 years ago, invented writing, and developed all of infinitesimal calculus, discovered the theory of relativity, and wrote it all down in a book that was kept secret, and by chance Newton, Leibniz, Cauchy, Darboux, Riemann, etc., as well as Einstein, read the book and attributed everything to themselves, or were part of a secret sect that forbade publishing the book. But that is far less likely.

In any case, that's not critical.
My questions are:
Can one rely on the antiquity of the Zohar? In my opinion, if such evidence were presented in any other field or subject, or in a religious court, it would not be accepted, and its source is dubious; and even if there is some doubt, it is nullified when there is an orderly tradition claiming the opposite, especially when the evidence is very questionable.
Also, the question arises: how does a book that no one had ever heard of and that was never mentioned anywhere suddenly appear?

In addition, what about the contradictions between Jewish law and Kabbalah? If both traditions are true, or have equal authority, how can a contradiction exist? Or more precisely, how does one decide between them? It is indeed accepted that we follow the revealed law, but there are quite a few who know this and still rule according to Kabbalah, and I have not seen people objecting to or protesting their rulings.

What is the attitude toward other interpretations, such as the rationalists like Maimonides? Does the Rabbi think their view is nullified before Kabbalah? How should one relate to them?

You are mixing together two questions. The antiquity of the book is not connected to the antiquity of its sources. Even if there was late editing, there are certainly ancient passages in it.
By the way, how many researchers do you think believe that the Five Books of the Torah are from Sinai? About the same number as those who believe in the antiquity of the Zohar.
There are contradictions and disputes within the revealed law too. So what?
No one is nullified before anyone else, because no one has authority. So there is nothing to nullify. Decide which interpretation seems right to you, and act accordingly.

Late editing might perhaps explain linguistic corruptions, but how does it explain things like an interpretation of the vowelization, when the vowel signs were invented in the ninth century and even then were different from today's vowelization? Or the factually incorrect description of the Land of Israel?

And let us say those are late additions and the rest of the text is ancient. How can one explain Amoraim who are mentioned as having sat with Rashbi or his students, when hundreds of years separated them? And when the whole study is with them?
That is impossible. Of course one can make claims such as: there was a quantum jump and the book arrived from another dimension where those sages lived together, or they went back in time, or Rashbi moved in the cave close to the speed of light and when he came out he reached the period of the Amoraim. Or that the dating in the Talmud is wrong and the Zohar is right.
But again, that is not plausible (just as in a religious court or a secular court, arguments like these would presumably not be accepted).

Or the mention of non-existent Tannaim who happen to resemble Greek philosophers in name, and happen to say things similar to those philosophers (usually with a slight corruption, or the text argues with their view).
How exactly do linguistic corruptions explain this or prove that the text predates the time of Rashbi?
Of course, without getting into apologetics.

Those interpretations, for the most part, do not claim information from Sinai or from the Tannaim. If we assume that the Zohar is from Sinai or from the Tannaim, then other interpretations cannot be accepted.
Of course, a person who does not accept Kabbalah will not be moved by that, but a person who does accept Kabbalah will see this as heresy. If the Rabbi accepts its antiquity, how can he accept new interpretations that contradict the Zohar?

Also, what does the Rabbi think about Besamim Rosh? There too there are early rulings, but it is accepted that the book is a forgery. That too could be legitimized with the same arguments.

I wrote that I do not see it as an authoritative source but as an interpretation. So why do I care whether it is early or late? You keep returning again and again to this irrelevant discussion.
And what does this have to do with Besamim Rosh? There it also does not matter at all that it is forged. Examine its arguments, accept what seems right to you, and reject the rest. What is the problem at all? That is why I also see no problem with such a forgery. I wrote here in the past that in my opinion this is permitted according to Jewish law (following the Magen Avraham and the Talmudic passage about hanging one's case on a tall tree).