Thanks for your thoughtful letter. This is a tough one because you are caught in the middle of something that you have very little control over. At 15 you are not going to be able to manage all the family dynamics. Whatever is going on for your mom isn’t something you created or can cure. But you can be clear about limits.
I would first talk to your high school counselor about what is happening. In order to deal with your mom you need some support, and the counselor is the first person you can talk to about this. Additionally, here is a free hotline for you to call that will put you in touch with a counselor who will guide you through the next steps to get support, and how to deal with mom when it happens again.
Your mother telling you to “fuck off” isn’t ok in any instance, and you need to have enough support around you so that the next time it happens you can confront her. Verbal abuse by a parent isn’t acceptable. Her blaming you and not seeing her role in this will mean you need support to help yourself. The counselor and the hotline should help.