Hmmm, I am thinking she has some kind of emotional/chemical/hormonal problems, and needs some professional help. It's not good for anyone when one person is upset *all the time,* and from what I've read at least it sounds like she's upset all the time (or 99% of the time). She needs some meds, or counseling, or both.
I can respect that you want to stay with your boyfriend, but remember that he is contributing to this problem too, by letting the girlfriend stay. He should put is foot down. I also like FullofLove1052's suggestion that maybe it would be good for you to have a place of your own to live, even if you don't break up with anyone. You could use more peace and less drama. Also like FullofLove1052 said, you do not have to keep trying to be involved with the girlfriend here, you can just stay involved with your boyfriend and leave it at that. This doesn't have to be a triad, it can be a V.
Love means never having to say, "Put down that meat cleaver!"