Could this mean "The dog is ours though"?
In Persian, we use "ki" at the end of our sentence to add "flavour".... in most cases it would be the equivalent of saying "though" at the end of an English sentence.
Yes, but you need to spell it as a separate word, and probably with a question mark at the end: "Köpek bizim ki?"
This sort of sentence is usually preceded by "iyi de": "İyi de, köpek bizim ki?" ("Okay but the dog is (already) ours, so what's the problem?" or something along those lines.)