There is no minimum age.
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.
Why would the father need supervision?
He will not be granted legal custody.
Generally, custody orders end at age eighteen and the child can choose where she wants to live.
Single mother has sole control and custody, regardless of age, at the time of the birth. Father has a child support obligation, only.
Any age. see links below
He doesn't. He goes through the appropriate court and will gain primary residential custody if it is in the best interest of the child. You do not "take" anything.
If this relates to custody, age 18. see links
If you live in the US... The father's parents have no rights to custody at all. The father has the right to petition for custody, but he won't be given sole custody unless he can prove the mother unfit. He might, however, be given joint custody.
If the child has not reached the state's age for emancipation, the custodial parent must agree to the change or the non-custodial parent must petition the court to obtain equal or sole custody rights.
Yes, harboring a runaway.
Yes, until the order is modified. If the child is over the age of majority, than it can go straight to the child. If not, he files for custody.