Is one allowed to ask a rabbi if he is Hassidic? Why or why not? Thank you.

Today it's no problem whatsoever, for if the answer is "yes", he is proud of his movement, and if not, he may just think that you are a "new-comer" and asking innocently. This is as opposed to before the Hassidic "movement" began about 250 years ago, where it would have been understood as if you're asking him if he is meticulously observant, or asking him if he is a "kabbalist" where, out of humility, it would have been uncomfortable for him to admit that he is on that high level. Also, kabbala is called "sod", the secrets of Torah, taught only to a selective few, which he would have probably preferred to keep secret. Mo'adim L'Simcha!