Luzy: Cutters have serious issues. This is a warning sign. You need to stay with counseling. What has been lost here was the rape you wrote about earlier. You mentioned this earlier, but understand that is another serious issue that I hope is discussed in counseling - preferably not online.
When you wrote the following "she starts cursing and screaming and then I tell her to calm down and she hits me and pull my hair ...", that's and indication that things are VERY seriously wrong on HER part. This shows that she can not cope with her emotions and is hurting you.
This behavior is not just a case of always arguing
and dealing with her mood swings. It sounds as though from what you write that she might need long-term professional help. It's very assuring to read that you're getting help now.
Her behavior towards your 'rule-breaking' behavior (being online and not doing homework) sound like way overkill. This behavior is damaging and could set a pattern for you and your future relationships with others, including boyfriends or, later on, husbands, should you marry. Being 'appropriately firm' and disciplining properly should never involve hair-pulling, etc.
Understand that saying 'Sorry' is a start to admitting there's a problem, but is not enough ... not even close to enough.
Yes, it's wise to do your homework, limit time on the computer, and stay out of her hair. Understand also, that she has some of her own homework
(counseling) to do too.