"Whose dog is it?"
Translation:Vems hund är det?
For questions, the rule is that the verb goes before the subject. If there's a question word or a phrase that functions the same way, that goes first. In this cases vems hund functions the same way as a question word would (compare: Who is it? Vem är det?).
After possessives, we use the indefinite, just like you do in English: 'his dog' hans hund (not "his the dog"), and vems works the same way – 'whose dog' (not "whose the dog") is vems hund. In vems är hunden, the dog is not defined (or 'owned') by vems (you can tell because there's a verb between them) so again it works like in English – you'd say 'whose is the dog' too.