"моей кошки" is shorter and more idiomatic, that;s why we have it as the main translation. Otherwise, both can be used.

I'm pretty sure it can be translated both ways. I don't know if there's a subtle difference.

I love my cat but sometimes he is a dick. Don't be lying that your one isn't from time to time

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).

Genitive

It is in the genitive. Alternatively, you might say "..., чем моя кошка"

It's the genitive singular form.