"Whose dress is it?"
Translation:Vems klänning är det?
I'd guess it's because you generally use "det" for anything not specified any further, meaning "it" or "that thing". So in this case you're asking "Whose dress is that [thing]?" rather than "Whose dress is that [dress]?".
I'm just trying to make sense of this, since I made the same mistake in thinking I should use "den", but I may be wrong..
Yes, that word order is a bit awkward. You could actually use it with a very marked question intonation, but then it would translate the sentence It's whose dress?? (with emphasis on 'whose'), not 'Whose dress is it?'
For questions with question words (like när, var, hur, vem 'when, where, how, who'), the question word normally comes first, just like in English.