I don't know the answers to your questions, but don't take your eyes off the ball.
You really don't care (or shouldn't care) about what happens to STBX. What matters is that the kid(s) are taken care of. Unless there's something really squirrelly going on, STBX's behavior will reflect poorly on her when it comes time for the judge to rule. No matter what happens wrt to the specific issue you raise, stbx will be looking bad from this.
If STBX doesn't pay support, turn it over for collection. CSE may 'offer' to rescind her driver's license, withhold tax refunds, etc and that may get her attention.