Whereas uninverted questions are normal in French, the Germanic languages all seem to work the same in this respect:
- Normal yes-or-no questions are inverted.
- Nevertheless you can take an arbitrary sentence and turn it into a question by raising your voice / putting a question mark at the end. This almost always has a special meaning, typically something like: "Apparently, ... . Is this really true?" This meaning rarely makes sense outside conversations, but good grammars still describe it as a standard feature of the respective language.
So, the translation of your sentence "Du har et dyr?" is "You have an animal?" This question would typically be the response to someone claiming to have an animal and would be followed by something like this:
- "Really? Since when? Didn't you say your parents forbid it?"
- "Are you sure? Didn't it die last week?"
- "Who would have thought it! I guess this explains why that object at the end of your leash is moving about."
As I said, this should work the same way in all Germanic languages, so normally inverted questions should be translated by inverted questions, and uninverted questions by uninverted questions.
You can say, "Du har et dyr, ikke?" Meaning you're not totally sure about your statement so you're turning into a question. In which they would respond if what you said is correct; "Jo, det har jeg." And now the opposite: if you wanted to say, "You don't have an animal...right?" You would say; "Du har ikke et dyr, vel?" And if they didn't they would say, "Nej, det har jeg ikke!"
Hope that helps! We went over this subject in my Danish language school in Copenhagen recently.