It can't be an accepted answer here because Har du ingen katt is a negative question but 'Do you have a(ny) cat' is a positive question – I'm talking about the difference between "having" and "not having".
The positive questions in Swedish would be Har du någon katt? or Har du en katt?
It's also problematic in itself to use 'any' like that with a single, countable, concrete noun in English. They normally only use any for plurals and non count nouns in this type of context. So if you wanted to ask that question in English, you should probably say 'Do you have a cat?' instead. In Swedish though, Har du någon katt? is a perfectly fine sentence.
There's a difference in English between:
Don't you have a cat? (= Assuming that the other person does own a cat and asking for confirmation)
You don't have a cat? / You have no cat? (= Assuming the other does NOT own a cat and checking if that's a correct assumption. Or an expression of surprise.)
How does this relate to the Swedish "Har du ingen katt?", which sounds to me like option 2 more than the given translation (option 1)?