Orthodox Judaism/Paul's claims
Q- Why does Paul (fron the New Testament) say that it was 430 years from the time of delivering the promise to Abraham to the time when the “Law” was given, when it actually was 645 years? Did he learn well under Gamliel or not?
you actually have 2 questions there... They can be answered with one answer though. First, did Paul learn well under Gamaliel or not, is the question we will tackle. Your question actually presumes that Paul learned under Gamaliel - which is a big "if" in itself. Indeed, Paul never mentions in his letters that he was a pupil of Gamaliel, even when he is most concerned to stress his qualifications as a Pharisee. This was actually a claim made by Luke as an embellishment to his narrative.
We are told by the Book of Acts that Paul, as a young man named Saul, left Tarsus, came to the Land of Israel and studied under the very prominent Gamaliel. Yet Gamaliel was a teacher of advanced studies, which means Paul would only have only been accepted to study under him if he had spent his entire childhood, and adulthood, deeply grounded in the rabbinical studies. This would have had to have been in Jerusalem, in one of the renowned academies there, which Paul himself contradicts by saying he just came from Tarsus. There are numerous other proofs we can unpack to show Paul was not at all who he claimed to be, and they're all outlined in an amazing book by Hyam Maccoby called "The Mythmaker: Paul and the Invention of Christianity" - it's available on Amazon and I highly, highly recommend you pick it up.
So... Paul never studied under Gamaliel at all, which is probably the least significant of the myths and embellishments he created in order to become, eventually, the founder of Christianity. Remember, Paul never met Jesus in the flesh, he claims to have received a vision on the road to Damascus (where he allegedly was sent by the High Priest who had no jurisdiction over the non-Jewish city of Damascus and couldn't possibly authorize arrests or extraditions). Nobody witnessed that revelation. Let me put it to you this way, as a parable:
There were 3 men whose father died and left behind a large estate. There was no will. One of the brothers woke up the morning after the funeral and said: "Father appeared to me in my dream and told me he is leaving all his money to me". Brothers rejected his statement and said: "If father wanted to leave you the estate, he would have appeared to us in our dreams, not to you in yours". Enough said.
So - before you count years between the promise of Abraham and the time the Torah was given and see if they match Paul's statements, you should consider the source. Paul was the maker of myths, and questioning his claims will be a very scenic route indeed.