No, your not related to her.
write your name
No, you can and should put the baby's fathers name on the birth certificate.
In this state the Mother can put any name she wishes on the birth certificate. It would be quite difficult to have your name removed from the birth certificate. It would be almost impossible to have the baby's name removed from the birth certificate. You could be Joe Smith and she could be Jane Miller. She could put the name Albert Einstein on the birth certificate and that would be the baby's name.
If you didn't sign birth certificate at hospital good luck. The only happy person that will come out of that situation is the lawyer.
The law varies from state to state and from country to country. In the United States, if you do not supply a name for the baby in enough time before the birth certificate is issued, the certificate will bear the first name "Baby Girl" or "Baby Boy" followed by the last name of the mother. The certificate can be amended later, when the parents decide which name to give the child. Until, then, the child's legal name is "Baby Girl" or "Baby Boy".
Like what?
If the babies mother wants the baby to have your cousins last name then the baby can. When the baby is registered if the baby's mother takes the baby on her own then only her name can appear on the birth certificate and the baby will take her surname. If your cousin wants his name on then he needs to go to get her registered as well and then the baby can have his surname.
The hospital will ask you the name, and when you tell them, they'll write it on the birth certificate.
no
Yes.
Whose birth certificate?
No, that would be against the law, the birth certificate is a federal document and must record the baby's mother's name. It is very kind of you to be looking after the mother and the baby, but do not break the law.