no the mother should have the legal obligation to take care of the child
if the mother terminates her rights can he collect child support from the mother if child lives with him?
Not voluntarily. The court must approve and the mother must not collect Welfare. see link
Child support would generally go to the person raising the child. Mother might even be ordered to pay to father.
1) to pay support; 2) to petition the courts for visitation
no
You can sign your rights away but you will still have to pay child support if you are the father or mother of the child. There is no way to avoid paying child support.
YOUR
last i knew unless there is a step father willing to adopt the courts will not let you give up rights if there is a step father willing to adopt you will not have to pay child support
No, you can't. First, you can't sue a parent for child support. Second, there is nobody to collect it from since your father has passed. However, if your father has an estate you could contact the attorney handling the estate or a private attorney to determine if you have any rights in his estate as an heir at law.
see related link
If he was not married to the mother, nothing until granted them by the courts. see links
An unmarried father cannot "choose" to not pay child support. The laws in every state require that a father pay for the support of his child. The mother must bring an action to the appropriate court so that a child support order can be established. Visitation rights are separate and a father can have visitation rights established by the court. Visitation rights are not dependent on paying child support.