Your sister isn’t drunk only on alcohol. She’s also drunk on power. By yelling and screaming, she gets to do what she wants when she wants. She reduces your mother and you to tears. Your mother (and father if he’s in the picture) need to take hold of this. Parents need to be in charge, not a rebellious teen. For some reason, your mom doesn’t seem to know how to set rules and limits and keep them. All your sister has to do is make some noise, and your mother caves in. Even though your sister seems to like defeating your mother and hurting the two of you, I’m guessing there is a part of her that is also scared of her own behavior. She doesn’t know how to stop it any more than you do.
Parenting an angry teen is tough. I’m sure your sister can be impressive in her anger. But, like it or not, she is still dependent on your parents. That gives parents leverage if they know how to use it wisely. I suggest that you show your letter and this response to your mother. Things won’t get better unless your mother learns how to gain respect from your sister. She may need to see a family therapist or attend a parent support group to learn the skills and to have the support she needs.
I wish you well.