By petitioning the family court in the jurisdiction where the child resides providing compelling reasons why rights should be terminated. You should be aware that relinquishing parental rights does not relinquish the obligation to pay child support or meeting other parental obligations, unless the child is being adopted by another adult willing to assume those responsibilities.
A father can gain custody by going back to court probably with an attorney. The judge will have to decide that the switch is in the best interest of the child.
Does she want custody or access?
He can still gain custody as the presumptive father.
yes
The father's going to be notified - as the father, he has that legal right.
yes
A child does not need to be abandoned by their mother for a father to be awarded full custody. If the father can demonstrate before a judge that the mother is unfit to parent, the judge can award him full custody of the child.
Yes you can gain custody of her.
see links below
The mother already has sole custody. The father has not rights. see link
Sole even if he was still shacking up with you
You file an emergency child in need of care and custody in court. See link to learn how.
Being in contempt of court can help the other party gain custody but does not automatically guarantee it. Many times a mediator or guardian has to be assigned to change custody. Attorneys and judges also get involved in this decision.