Responsibilities of grandparents towards their minor's child?

Are people legally obligated to care for their child’s child? Would the baby be able to be placed on the grandparent’s health insurance? Would the parents be able to keep custody of their own child, and opt out of caring for their grandchild?

In the case of very young parents (say, twelve year olds) would they have the right to medical decisions regarding their child, if the grandparent disagreed? Would a young girl have final say-so over her body during a pregnancy? I know a girl cannot be forced to have an abortion, or to carry a pregnancy to term, but what about amniocentesis or other tests?

What rights and responsibilities would the father’s family have?

I know the answers will vary depending on the area, but this question isn’t based on a real life situation, so the information doesn’t have to apply to NYS. I’m just wondering how these things are handled in general.

In many cases the grandchild could be put on the grandparents health insurance. Typically, if the parent qualifies as a dependent then the grandchild does as well.