Look, I think it's commendable that you've moved in a positive direction with your children. Hopefully, you will be able to continue to controll your impulsive, destructive behavior, if for no other reason then the sake of your children who look up to you because you're their dad......pehaps some anger management classes can help you channel your anger into something constructive not destructive (holes in walls and doors, breaking things, etc)
Question: You said you worked hard.....do you have health insurance and, if so, are your children included on your policy?
Regarding Medicaid: Continuing to tell them you do not live in the household is perpetuating the lie your GF told them....such a wonderful example for the kids to look up to.
If you repay Medicaid, it will come out of your paycheck until paid off and, yes, they will consider your current monthly expenses.
and lastly, we ALL pay taxes but that doesn't give you the right to try to cheat the system.