"моей кошки" is shorter and more idiomatic, that;s why we have it as the main translation. Otherwise, both can be used.
Exactly. It is an irregular comparative, which is similar to how English "worse" does not come from "bad".
In Russian, the form хуже comes from худой, which in the modern language means "thin, scrawny" and does not form this comparative anymore (худее is its modern comparative).
It is in the genitive. Alternatively, you might say "..., чем моя кошка"
I had a cat that was horrible and very selfish. But I still love it like a serf. It died in my sister's arm. That was more than ten years ago now... :*(