possibly adopting a group of 3 brothers through foster care

This is tough. I say this because it's hard to keep siblings in touch as they get older. If you do take them, they need to be in trauma therapy. EMDR is one; there is another one but I'm having a senior moment. At 13, he definitely should have chores. See if the social worker will help you get them in trauma therapy. That should help with the behaviors. Then you can work on school, chores and other stuff. Good luck!