My girlfriend is mentally, verbally and emotionally abused at home, how can she get out of her household?

Try to talk her into calling the police, because this would be the best thing. If she won't, try talking to a trusted teacher or counselor at school. Even though it's the summer, you can email them, and maybe they'd be able to help. Do try to get her to stop cutting, because, as a previous poster said, it could end up being fatal. Good luck. I hope everything works out.