Does "ziek" refer to general illness like "sick" in American English, or physically sick (vomiting) as in British English, or both?
It is unwell, ill. To vomit is overgeven, and you would ask, 'Overgeef je?' (Are you vomiting? but in a rather direct, 'do you vomit' way) or 'Ben je aan het overgeven?" (Are you vomiting? in a bit less direct way)
I'm Belgian. We do say overgeef je as well as geef je over. Both are good.
Does this only give a literal translation of the English sentence? What if one says "Are you sick?" in the sense "Are you an idiot?" - can it be as well expressed by "ben je ziek" in Dutch?
Sure, you can say that in that way, but of course, it needs to be in the right context. Not so much to mean, are you an idiot, but rather to mean, are you sick in the head. I would imagine a scenario like the one person talks about putting something unpleasant in another person's drink, and the third person then says, 'ben je ziek?', like English would say, 'are you sick?' But you'd more likely ask, 'ben je gek?/are you crazy?'.