@Ticomaya,
The state or states in the US just wish normally to have some male helping pay the freight in the raising of a child and care little about being "fair".
If the child is born during a marry or 9 months after it ended the husband or ex-husband is for the most part on the hook and this go double if he did not challenge his fatherhood in the first few months after birth.
See a lawyer as the laws are in some flux over this matter and is not the same from state to state.
As far as the child name that is not the property of the man and no court is going to order you to change it, so he get to keep his name if you wish him to.