Can a unmarried non custodial parent voluntarily give child support and does that give the noncustodial parent certain rights?

Do it right and get a court order. Leaving things informal is great until there is a disagreement and then you'll wish you had orders to fall back on. And, like home repairs or anything else, it will cost more later to fix things than to do it properly the first time.

Voluntary payments don't give you rights. Your questions clearly shows that there are a number of legal issues including, child support, visitation, parental rights and possibly paternity. Speak with family law attorney and review your options.

I am not licensed in TX so you need to check with a local family law attorney. Paying support may obligate you to continue to pay whatever that amount is, even if it is more than the court may require; and, it gives you no rights. As the other attorneys have said, find a local family law attorney and get orders in place, doing it right the first time.