You need to set limits with your friend as whilst you have no one to 'off load' what she is saying onto it is becoming a serious issue for you. Whilst it is good for her to feel that she can talk to you, you can't allow her to be upsetting you in a great way. You're only 16 and I don't think you're in the position to play the role of the counsellor as well as friend, its not your job to. Counsellors are trained to be able to 'turn off', friends aren't.
I don't think you should end your friendship with the girl. Yeah, she has problems and she's abit confused but she needs you. However, you need to make sure she knows you have clear limits within the friendship- she can't depend on you and tell you 'everything' as you're not qualified to help her and its worrying you. That said, I think you need to talk to the teacher she speaks with at school and mention that you're worried as she is cutting herself in school in front of you. Thats plain manipulation.You're not breaking your friends confidence by telling a teacher as it will help her. If the staff already know about her then its ok, tell her that you're telling the people who can help her properly. Its not a betrayal of trust.
The fact that she is cutting around you is the most worrying... maybe she feels she needs some form of validation for what she is doing? A cry for help and to show someone how much she is hurting. Sort of, she 'shows' you and then because someone else has seen her do it, everything becomes real in her mind, whereas when you're on your own it can become very easy to think 'what I'm doing isn't dangerous, I'm not really a self harmer, its ok what I do, its not bad enough to be serious' when in actual fact, if you're cutting yourself, you do have a serious problem. Personally, I would never have let anyone else see me hurt myself as its a very private thing and I couldn't even comprehend doing it infront of someone as it would hurt them... but I can sort of see why she might feel the need to.
Just for the record though, I doubt she would attempt to cut you, or anyone else. Like BellaCat said- it's a very private thing, and her knife/blade/whatever she uses will most likely signify alot more than just an object to her. When I was cutting my knife became a part of me and I would never have used it to hurt anyone else, or anything bar myself. It was the only thing I ever used, then I lost it, and I can honestly say I was upset for ages...it felt like because I had lost that, I'd lost the things which kept me alive... which is why I wouldn't 'share' it with anyone.
I admire you, its a tough job putting up with someone with 'problems'. Alot of my friends deserted me, and then I began the some of the few who stayed with me (long story...) and yeah, its tough. I've been on both sides, as the friend and as the self harmer... and I think its hell being the friend as all you want is for the other one to be ok again.