yes, she has sole custody
She can terminate her parental rights, not yours.
The mother is the legal guardian from birth and can give the child any name she wants but she can not sign for the father on the birth certificate. He have to sign it and if she was below age of consent when the child was conceived he can go to prison for statutory rape.
yes
File a motion to modify custody. see links
The context is California law
You might. There are two different concepts going on here. One is parental rights; you can sign those away because they're yours. The other is the child's rights. You can't sign those away because they're not yours, and the child can't sign them away because as a minor he or she is legally incompetent. One of the child's rights is the right to support. If the child's mother is supporting the child fine on her own, she's not legally obligated to seek support from you, so you might not ever have to pay support. If the child's mother's situation changes so that she's no longer able to support the child on her own, you might wind up having to pay child support. If she's on any form of government assistance, the government may insist upon it.
What exactly are you asking? If the mother is married the husband is automatically the father legally unless the biological father sign the birth certificate or prove paternity in court.
If the father is considered legally to be psychologically incapable (or impaired) to be a fit parent via evidence presented by a forensic psychiatrist in a criminal case, usu. held at either the district or state-level judicial court; then, yes, the father is must sign all legal rights to the child's mother.
If you sign a prenuptial agreement then you are legally bound by that agreement. What did you agree to?
Legally ii is extremely difficult to legally abandon your responsibilities to your own child. Insufficient information is disclosed. Sign over your rights to WHO, WHY, and for WHAT reason?
Yes they can and so can the father
No, she can only sign over her own rights.