Yes
The parents have assumed joint custody.
If you only want to divorce them (and aren't worried about dividing property or child custody, support, etc), you may be able to serve them by publication and get the divorce. Probably best to talk to a lawyer in your area.
Mother files motion for custody
Yes, but child custody can not be finalized.
Will you have custody or not?
Only NC has jurisdiction, nor should the child be taken away from the other parent.
When you get a divorce ,if your spouse has moved in with you ,you can ask them to move out and if you have had a child with your ex-spouse your child lives with you or you can have shared custody of the child but if it is a teenager when you and your previous partner get a divorce you will ask them to choose who they will live with and they can either choose you or your ex-spouse or shared custody
A child is not allowed to choose until he is 18 but in a few states a child 14+ can be asked when custody is being decided regarding divorce. The judge is not always obligated to follow the child's wish.
Ordinarily, a trial court will enter "temporary orders" for custody, visitation, and child support, while the divorce is pending.
Unless the divorce decree states that they do, no, not unless the person having full custody permits it.
A divorce with the custody of a child involved can get very tricky when one parent lives abroad. It is almost impossible to force the absent parent to pay child support, and if the parent takes the child to some countries it may be impossible for the US parent to get the child back,
Custody refers to legal guardianship. It determines who a child lives with, who can make decisions regarding the child for legal purposes, and under what circumstances they can make those decisions.