"She has no dog."
Translation:Hon har ingen hund.
it says "Hon har inte nån hund" when I make a mistake. I assume it is one of the possible options here? If so, does having two translations mean that one of them can be used when I specifically want to refer to the dog as fellow kin?
inte någon, or with a very informal spelling, inte nån means 'not any' in English. Now, in English it isn't idiomatic to say "she doesn't have any dog", so you'd just say a dog instead, but in Swedish, hon har inte någon hund is perfectly natural. In the plural, it works the same again: hon har inte några hundar is she doesn't have any dogs.