If the current custodial parent is not willing to relinquish (or take), custody, then you have to retain an attorney and have it taken before a judge.
You file for it in the jurisdiction where the child legally resides.
Yes.
No, you slowly make it harder, i.e. first use half, if father continues declining, file full custody
Is he needing custody of his child, or is this on himself?
A motion for custody or modification to an existing custody order must be filed in the jurisdiction where the child legally resides.
Yes, but child custody can not be finalized.
file for sole custody and file for child abuse and you will have possibly sole custody of thew child
Yes, if you had custody of the child during the period for which you are seeking retroactive support.
Only NC has jurisdiction, nor should the child be taken away from the other parent.
You have the right to file for a change of custody with the court. You will have to present convincing evidence that the child's mother is unfit to retain custody of the child. Frankly: It is a stiff burden of proof to overcome to convince the court to remove a child from its mother's custody.
See Link BelowChild Custody- Can Fathers Win
Filing for child support has no bearing on the issue of custody. Custody is decided on the basis of the welfare of the child. To get custody of a child, the father would have to demonstrate that he would do a better job as parent. I will add that even if you do need financial assistance, that does not in any way suggest that you are not a good mother. Those are separate issues.