Actually I am not an attorney.
It does not matter what "underlying" issues there are, if he is the legal father of this child, the courts do not allow him to walk away from his responsibilities.
What DNA test was used? Was it a lab or a home one? Anyways as long as their is no legal father of this child, your friend has no rights, therefore there are no rights to sign away unless he is already signed the birth paperwork for the child. Is there a father listed at all? From your other post you mentioned a birth cert. Did your friend sign an acknowledgement of paternity? If he did, he is the legal father it sounds like and he should set up some legal custody/visitation status through the courts.
In regards to the latest post, your friend should not be paying this woman ANYTHING without going through he local child support bureau. If he continues to pay her volunatarily, he needs to write on the checks or money order, child support otherwise mom can file for back support and claim Dad gave her no money. He need to cover himself.
Secondly, mom's little request of having sign a piece of paper is nonsense. He does not have to listen to her and in the event something does happen to mom, he is free to try to legally establish custody. He should not be thinking of the "what if's" at all. He is either Dad to this child, or he's not. He cannot legally terminate his rights to this child. If he wants to voluntarily pay support to mom and have no rights to the child at all, thats his choice.
OR, he can leave it the way it is, and hope that mom does not file for any sort of child support, although she can at any time, even 10 years from now.
If he is the legal father and he signed the birth paperwork, he's dad, and he should look into his rights a father. The courts will not allow him to terminate his parental rights. He would be foolish to listen to mom and sign any papers mom might give him.
He needs to grow a backbone to mom.