That completely depends on your state of residence (or country if not in the USA) and the age of majority for that location.
All states have statutes of limitations outlining how long a parent has to file for and recover any back child support if an order wasn't filed when the child was a minor. If such an order was filed, then yes, the father may direct the child support enforcement agency in his state of residence to collect any support in arrears.
yes he should do
Sole custody means you have all the power. You can choose to keep the father out of their life until the children turn 18 if you want to or you can allow visits.
you get it for her.
Yes, if the father is not given custody he will be obligated to keep paying support to whomever the court awards custody or guardianship of the children.
yes
My answer to that would be 'No'. The father is responsible for providing child-support regardless of who has custody of the child; at least until the age of 18.
The father would file for a visitation order, not child support. Generally, if the parents are unmarried the mother has sole custody and control in most states until the father can establish his paternity. Remember, a child's mother can always be identified by medical records. Since the father didn't give birth and he was not legally married at the time of the birth he can establish his paternity through a DNA test. A paternity test can be arranged through the court. Once paternity is established in court, the father can request visitations or custody through the court. If the mother retains physical custody she can request that the court issue a child support order. If the father gets physical custody he can request a child support order.
How does he have any visitation rights with a custody and child support order?
You may have to pay child support to the one who has custody of your children.
If you're the father
yes
If you have Joint Legal Custody, then neither parent can physically change residences without the approval of the other. If, per the question, the daughter is living with the father, or the father has "primary" custody, then you can move wherever "you" want, at least in my perception.