Well first off "себя" means "oneself/myself/yourself" on it's own, so there is no need for "твоя". And убрать needs to be in the imperative, so it should be "убирай-те". Since себя is an animate object, and it is the direct object, it also needs to be in the genitive case. Which it is. (The nominative cases would be сам, сама, само. Dative - себе. Conjugates mostly like тебя.)

это кошка Алины. This literally means "This is cat of Alina. Alina is in the genitive case because it's describing possession. If you were saying, "This is a cat" you would just say "Это кошка".

У меня есть бирюзовой машины. "бирюзовой машины" shouldn't be in the genitive case. In this case, you would only need to use the genitive case to describe negation, or the absence of something ex: У меня нет бирюзовой машины.

I'm not a native speaker, so feel free to correct me if I'm wrong.