Child support and custody/visitation are separate issues. While the court won't look kindly on him being behind, they won't necessarily be more favorable toward you. The important thing to remember is that the goal of family court is to look out for the best interest of the child.
As for him moving, if you were able to contact him, then it would be nothing more than a slap on the wrist with an admonishment to follow the orders next time. Not likely to work in your favor.
The fact that she stays with his girlfriend is irrelevant unless you can demonstrate that she is an unfit caregiver. He has the right to secure childcare when he has posession.
Have you recently read your custody and visitation orders again? Mine are worded so that I can move within a certain radius of my current residence without any discussion. I only need to notify him of the move. I want to say it is 150 miles (but I don't have them in front of me).
I wouldn't move and then wait to see if he contests it...that will just take more time and money for everyone. Try to agree together on a plan and then take it to court for approval. Given that you will be getting married, and thus creating what will be likely considered a more stable environment, I can't imagine it would be a problem. However, the judge may want you to be married first!