Father has not seen the child since Jan 07. He is 3500 in arrears, and mother wants to file abandonment and TPR to prevent future visitation (father is drug addict and history of violence). How will TPR affect back child support and any future child support?

A TPR has nothing to do with support unless it is the first document filed in a much bigger case.

Example: Your wanting to get divorced and there are domestic violence issues. In the TPR it might have other orders in it also.

Since support is already set then it has nothing to do with it at this time.

The other party may file to be allowed supervised visits, also is there a history of violence to the child ?