First question:
Yes. Think of it this way: "remove yourself from where?" "from my house".
Also, «убрать» needs to be in the imperative form: «убери» (if you were on «вы» terms with your interlocutor, you would add «те» to the end of imperative forms: «уберите»).
I believe the correct sentence would be: «пожалуйста, убери себя от моего дома сейчас»

Second question:
«это кошка Алины» is correct. "Whose cat?" "Alina's cat". «кошка» does not need to be in the genitive.
If it was «это кошки Алина», you would be saying "this is the cat's Alina". If genitive is being used to show possession, it is the agent that possesses that requires the genitive, not the object being possessed. In English, we simply use the possessive "s" for this (Alina's cat).

Third question:
The sentence is wrong. «я» is the agent that possesses the car, the genitive of «я» is «меня». You should not put the object of possession into the genitive as well. Your sentence should be «У меня есть бирюзовая машина».