Typically, a parenting plan will outline visitation beginning with a phrase along the lines of "During the term of this Parenting Plan the non-custodial parent shall have at a minimum the following rights of parenting time." If so, the non-custodial parent is not required to exercise his or her visitation time, but is allowed to do so.
Extracurricular activities can be fairly minimal, or can be thousands of dollars per year, per child - especially if the word "travel" is involved. Because the costs are usually on the low end for younger children, parents often don't consider the costs at the time of separation, but only when they've reached their full, teenage levels.
I'm frequently asked by parents who pay child support if they can demand an accounting from their ex of how exactly the child support money is being spent, as they don't believe it's going to the children (but rather a car, house, etc). This idea is the basis of the song "Gold Digger".
When child support is ordered, either after a hearing or by agreement of the parties, is has to be paid. But often, people are unaware of how this is enforced. There are a few ways.
There are two issues surrounding where a case will be filed - jurisdiction and venue. There is a course in law school dedicated almost entirely to these issues (civil procedure), so unfortunately I can't give a comprehensive answer to the question that suits every situation. The general rule is that cases must be filed in the state and county where the person you're filing against lives, but divorce and family law have more exceptions to that rule than probably any other area of law.
One issue that comes up frequently regarding custody issues from unwed parents - whether a father asking what his rights are, or a mother asking about her responsibilities to the father - is that, though the father hasn't legitimated the child, "his name is on the birth certificate." People often - mistakenly - believe that this means the father is the father for all purposes in the eyes of the law.
I understand this confusion - you've got one of the most official documents in a person's life, issued by the state, naming the father of the child. It seems like this would be the state's recognition that the father is in fact the father. But it's not.