Ultimately this is a question for an attorney.
However, in general, the court will look at the specific circumstances and evaluate which parent is in the best position to provide a healthy environment. The general rule is that "the best interest of the children" is the overriding factor.
In an emotional, contested action each side often presents conflicting "facts" and "evidence" to indicate that they are better suited, and the other side is unfit, to have the kids.
In those cases, the court must evaluate the information and ultimately decide "The Truth."
Ultimately, the parents don't decide that - the court does, taking into consideration the basic character of the two parents.
Arranged Marriage
If you mean a marriage where parents decide their partners for their children, it's called an arranged marriage.
Would not take place as both signatures are required. It depends on whether the parents share legal custody (separate from physical custody) and if the state where you live requires one or both parents signatures. Some states will allow the judge to decide if the marriage should take place in the event of pregnancy and/or birth of a child to the couple.
Good question. The likelihood is that Social Services will get involved and decide which set of parents is most suitable for them. If both sets are suitable then it may go to court for a decision.
Probate Court
Usually the Court would decide that.
Minors can not decide who to live with. That is up to the parents or the judge. In some states the judge can hear what the child wants but is in no way obligated to follow that wish. The children can decide where to live when they are 18.
if you want
The non-custodial parent can go to court and request custody of the children. A judge will decide which parent should be awarded custody of the children. The circumstances under which the children are currently living, combined with the reason(s) they are not living with you will be a big factor in whether or not you get to keep custody of the children.
If they have decided so, yes. It's the parents who decide, not the minor.
No, there is still a parent left with custody. And custody can never be willed. That is for the court to decide. The ones in the will can ask for custody but it is up to the court.
Of course not. Mom and Dad (assuming they have custody) get to decide where you live.