If a parent decides to terminate parental rights, how are child support judgements and orders past and future handled?

The same way they were if rights were not terminated. Terminating rights does not mean terminating child support. Basically it means that the parent who terminated has no right in any decisions, medical or otherwise, as to how the child is raised.

Financial responsibility is rarely terminated.

Unless the other parent has a partner that is going to adopt, giving up the rights on a child does not change or stop child support, it is figured and inforced the same way as with those that have rights.