"Kot" and "pies" are in Nominative case and "kotem" and "psem" are in Instrumental. "Psa" is Genitive. Although it might seem bit strange for a non-native speaker, we use Instrumental case with verb być (to be) talking for example about your/someone else's job/nationality/identity. There is some explanation about Instrumental case here: https://www.duolingo.com/skill/pl/Defining.
The only situation when negation changes the case (but a very common situation), is when a verb that took a direct object in Accusative gets negated - it takes Genitive instead.
Other cases (like here you have Instrumental) stay unchanged.
You can read more about it here: https://forum.duolingo.com/comment/28545847