why dont the children want to stay with their mother?
"sometimes he has to bribe his little brother to go stay with him at his moms house"
do the two boys just feel happier and safer with you? Talk to them bout it and find out why they dont want to stay with their mom. Even if she is doing ok, if they are just happier children with you then i think you have your answer.....
PS the selfharming (cutting and pinching) is very self destructive behaviour - it should be looked into.
She should be given the benefit of the doubt unless you are aware that she is abusive or neglectful to them, perhaps they simply are getting used to the situation, the 8 yr old could use some counseling for his behavior to himself,ask his Mom about that , you say she is trying and it seems she is doingwell with them, how about you support her and give them positive feedback about her.