In principle, questioning is a virtue in Judaism. The idea is that if you haven't examined what you believe and had it challenged, how can you be sure you believe it? That said, some (quite a few) of the orthodox sects have this thing going where if your questions don't come up with answers in the right vicinity there's something implicitly wrong with you, which seems like extensive pointmissing to me.
It's possible to convert to Judaism but for many purposes a first-generation convert isn't considered "fully Jewish". This won't affect your afterlife prospects or your relationship with God or anything like that, but for things like funeral funds and Israeli citizenship you often won't qualify. The idea, like with the rabbi trying to dissuade you, is to keep people from converting for reasons other than an actual desire to embrace the Jewish faith. That said, some people will hold various kinds of prejudice against converts, but what can you do about dicks.
Belated l'shanah tovah to y'all too. I was going to make some kind of shofar/fo'sure inversion pun here but I couldn't think of one.