Probably not. At least, if she gave birth in the US, that can be an ordeal - keeping a spouse off a birth certificate initially might not be hard but he can be added easily. He is also the presumed father since they are married, even if he isn't named on the birth certificate, and would have rights and responsibilities such as child support. At least, that is my limited understanding. That probably varies by state though.
This is correct. He is the presumed father with parental rights and responsibilities.
Grandparents' rights were addressed in the Supreme Court case Troxel v Granville. Grandparents can intervene in a pending custody dispute for access rights, like in the case of a divorce. This doesn't necessarily mean visitation, but some is likely to be awarded if they have had a relationship with the child. What a judge decides in a case like this is anyone's guess but visitation would take place where the child lives. There may be variations in each state, but many adopted the basic thought that a fit parent has the right to choose who, when, how their child interacts with others. The courts don't feel it's their place to override a fit parent's judgement in raising their children and it's presumed they are fit unless proven otherwise in court.
A judge won't orphan a child just because the parent is a jerk. Someone else MUST step in as the parent if one agrees to relinquish rights. This line of thought is a moot point. It's not going to happen with the present circumstances.