Hey, I get ya. I would not like it if someone just start cooking in MY kitchen. I never would violate another woman's kitchen like that. Even when we stayed at my Grandmothers, if I wanted to give her a break and cook the breakfast--or whatever--I would 1st ask her if it was ok. Especially since you already put out there, to everyone, how you feel about your kitchen, what your dad's partner did was a gross disrespect.
The "not ever" hang out with him was a bit too harsh, but I can see it was out of frustration. I think you should find a moment to apologize to your 10 year old brother--he is just a kid--let him know you love him and will be happy to play with him when you are done studying, but right now you have to focus. Then go to Mom and let her know that you need her to find something for him to do, so you can do what you NEED to do. Good luck.