Yes. "House" would be better since it refers to the physical object whereas "home" expresses the functional meaning of the word, the same respectively for "hus" and "hjem". In conclusion I believe there should be "hus" in the danish sentence, but this is a learning program and these slight semantic differences should be accepted.
These are the English ways to use "have" to express ownership that I'm familiar with:
- Have you got a car?
- Do you have a car?
This is the thing that makes me cringe:
- Have you a car?
I do think that "to be" is the only verb that can used without an auxiliary when forming a question or a negation. With "have" you either need "do" (as auxiliary) or "got" (as a present perfect construction).
English is the only language I know that requires auxiliary verbs in questions and negative statements. You're learning a lot of languages and I think you can confirm this.
Anyway, as long as I've been taught English (British English, mind you), a question with "have" as the full verb and no auxiliary has never come up, so it just sounds odd to me. But if you think it's alright, go ahead and suggest it. I'm not a monster. :)
(And I really won't complain if you're going to be very classy and go for "Hast thou an automobile?")