1: Yeshu mentioned in the gemara couldn’t possibly be oso ish the xtians are celebrating. Yeshu of the gemara was a talmid of yehoshua ben prachya (according to the gemara) who predated hillel and shammai by several doros (see pirkei avos), while oso ish the xtians celebrate was supposedly a contenporary of Raban Gamliel and a talmid of hillel according to their scriptures.
2: The xtians themselves have surely made up a birth date for oso ish, just as surely as they made up his yichus (different books of xtial gospel list very different yichus back to david hamelech – with one listing 24 generations and another more than 40 – with most of the names different in each – and one listing the yichus through the mother, since they believe he had no physical father). The birthday chosen is one that coincides with numerous old pagan beliefs.
3: To learn or not learn on the night of nittel is surely a minhag that some have while others do not. Instead of haggling over yes or no, it makes way more sense for us to focus on how to reach out to the vast majority of jews who will not learn a word of torah, unfortunately, on any night of the year, and worse, might have been celebrating his birthday on that night.