If I start child support voluntarily can I have a say in where they live or move?

You have an obligation to pay child support. It has no impact on your custody or visitation rights. If you do not have a court order that gives you custody or visitation rights, you need to get one. She will be able to move within the state, but you will have a set visitation schedule that must be followed and since she chose to move so far away, she may be responsible to provide transportation for your visitation to occur. Your best option is to at least consult with an attorney before you file anything in the court system.

While the amount of custody each parent has helps determine the amount of child support (as well as obviously the incomes and credits for health insurance/child care payments), the two claims are completely independent. So just because you pay child support doesn't automatically mean you have joint custody (physical or legal) and would have to actually file a lawsuit to start a custody action to get that resolved. On a related note, if parent A denies parent B custody, that doesn't give parent B the right to stop paying child support. Likewise if parent B stops/misses a child support payment, that doesn't give parent A the right to deny parent B custody. You should consult with an attorney to discuss if you should be filing for child support and custody together and help in doing that so you can hopefully present the best case and get some reasonable custody/visitation along with the child support.

No. You should be supporting your child because it is your right and responsibility. However you do not say whether you have established paternity or have any court determination. Without that she can move wherever she wants.

Rather than simply agreeing to provide support off the record privately, you would be better served to initiate a paternity action to confirm your status as the father, and seek to formally establish visitation and/or custody parameters as well as support amounts.