A child doesn't come under the jurisdiction of the court unless there had been a divorce. If you were legally divorced you would need the court's permission to take the child out of state. If you are not under any court's jurisdiction at this time then you are free to move.
If married, both parents. If single, the mother
Not without the permission of the child's biological mother. When a couple are not married and there is not a custodial order from the court, the law presumes that the mother has sole custody of the child in question.
mother has sole custody even if living with father
Yes. A father has no automatic right to their child (unfortunately) unless the parents are married.
Doubtful. The child support is for the welfare of the child, not the mother. Check with your state child support office for specifics of the law in your state.
Mother has sole custody and control in 49 states see links
In most states that default is the mother. The court can determine otherwise.
Married parents have equal parental rights. File an injunction to have the child returned to the state as soon as possible.
It depends, are you divorced from her. If you are, the answer is no if you are planning on staying there. However, if you are just visiting the state, then there should be no problem. If you and the mother have not ever been married, then there is nothing wrong with you taking the child.
Once a child is in state custody I don't think the mother has a say in who adopts the child. However, you may be allowed to meet the adoptive parents.
Yeah, send them to a realtives house.
If they are legally married, the father gets rights until mother gets out of prison, after that it is up to the state. If not legally married, they go into state custody.