Can I say: Ten kot lubi nas?
Yes, this puts emphasis on 'nas'.
but when I wrote "ta kobieta lubi je" it was wrong, why?
We also wouldn't accept "Ten kot lubi nas". We just don't consider that emphasis natural at all, you rarely mean "This cat likes US!" This woman likes THEM!".
So "nas" = us, and "nasz" = our?
Why is nas genitive and not accusative? sorry I have so many questions!
"nas" is Accusative here, as it's simply the direct object of the verb "lubi".
Although Genitive would be identical.