"Is it your own dog?"
Translation:Är det din egen hund?
det doesn't refer to the dog. It's a placeholder pronoun, or rather, it refers to "the thing we're talking about". If we use a pronoun about pets, we normally use han or hon.
Long post about this here: https://www.duolingo.com/comment/9708920 There's an example in there about how if there's a knock at the door and you know who's there before seeing them, you would say It is my wife, not She is my wife. I think that comparison might be helpful in order to see what's going on in this sentence, too.