This would depend on the Will. If not addressed, were the parents married? If not, the paternal grandparents have not claim as unmarried fathers have no assumed rights. Otherwise, if the parents were married, and no Will exist, than it would be a matter for the Probate court.
No it is not mandatory. Some parents (could be the bride to be's parents) may have the groom, bride, their parents and the wedding party out for dinner in the evening a few days before the wedding, but this is not mandatory either.
Parents that are married to each other have equal rights to their children and does not have to ask permission from the other one. If they choose to have some rules within their marriage it has nothing to do with the law.
I have this same issue. My parents never got married. My dad's right was to see me, so he did not have legal custody of me. It varies with different families, but the father has legal custody to see you and it varies if he can have custody to have you.
Missing way too much information to give you a real answer. Let's start with the age of the parties involved. If they are under 18 and are not married, they are still minors and subject to direction and control by their parents.
You can get married at 16 in most states with parental permission. The requirements for proof of permission vary from place to place. You will have to consult your local licensing authority for what they require.
The baby's last name is irrelevant - in the US, parents can give their children any name. Before support is ordered, paternity would need to be established through marriage of the parents when the child was conceived or born, acknowledgment, or genetic testing.
When the parents are unmarried the mother has sole custody until the father establishes his paternity legally in court. Once established he can request joint custody and/or a visitation schedule. He will also be subject to a child support order.