Привіт! If I say "I'll go home", will I say дім, вдома or other version of this word?
whereto? to one's place - додому, to some house - до дому
They would, but it's not a problem. "до дому" exists so that you could insert an adjective here: Я йду до старого/нового/батькового дому :)
Not really. Of course, we can imagine a situation like you are leaving some place and say: "я буду вдома," such that everyone understands that you are going home right now. But you do not say that directly.