No. Both parents permission is necessary.
Yes, but the father can file an injunction ordering the child be returned to the jurisdiction of the courts. Better to get it approved. see link.
It depends how old she is and she should at least let him know
If the father doesn't have joint custody, then yes. If he does have some sort of custody, probably not. If you went to court, look on the papers or call the court and see how much parental rights he has.
only if the mother has full custody or if they are married i am pretty sure that is legal.
There is no requirement to have approval. As they are not married, the mother has custody. Where she and her child live is up to her.
I don't believe the mother can do that in any state. There is an order of how things are supposed to go. The mother must ask the courts permission to go and it must be granted. Fathers do have rights.
If the mother is unmarried and the father has never established his paternity that might be possible if the mother claims the father is unknown. If the father knows he has fathered a child he can request a DNA test to establish his paternity through the court. The child could not be adopted without his or the court's consent
I'd say she can unless ordered not to do so.
If he has primary custody or even visitation rights, you cannot take his child far enough away that he cannot readily exercise his custody/visitation rights unless he gives you permission to do so.