So "Van ons" translates literally to 'from us'? If so, "De hond is van ons" translates literally to 'The dog is from us', can be understand as 'the dog is a part of us' and therfore it is ours?
Most likely also 'De hond is van ons', but "the dog is from us" doesn't really sound like the preferable way of saying this in English. "The dog is ours" sounds better (but I'm not a native English speaker so correct me if I'm wrong).
If you were, say, giving a dog as a gift, "the dog is from us" it would work. But you're right, in normal, everyday English if you wanted to indicate ownership of the dog you would just say "the dog is ours"