So, Child Support in any state is a very convoluted backwards kind of thing. The laws may be different from state to state, and state to state it varies as to how child support is calculated. However, here are some basics.
1. The dominant parent does not have to change the child's last name when paternity (if relevant) is established. It is not a HAVE to kind of thing. It is a choice, as in you give me this and I'll give you that.
2. The tweakings and concessions made in regards to child support are NOT done through the child support agency. Instead, they are done through lawyers. So if you're dealing with a jerk of a dead beat mom or dad, you may have to get a lawyer involved.
3. It is not possible to have your cake and eat it too. I've been told in the past by other parents who have dealt with child support that you can get support while also having the other parent terminate his/her rights. This is not true. You can, however, hire a lawyer and set up an agreement saying something to the effect of: I will accept this lesser dollar amount as support over x number of years if you waive parental rights.
No doubt I'll have further illumination as the process continues and I learn more about what's really going on. But I will leave you with this little suggestion...
Should you ever need to open a child support case and you feel that nothing is getting done, do not toy with supervisors and the like at the child support agency. Instead, write a letter to your governor (not your mayor) giving them all of the pertinent information of the case as well as how/why you feel things are not being done. In many cases the governor will get on the case of said agency and force them to do work for their money. Good luck.