I am the mother and have custody right now. There has been no paperwork saying I cannot leave the state issued. What will happen if I miss the date?

If there is pending litigation and you don't show up a warrant can be issued for your arrest.

What is the case about, If you move out of state, the other party can file for temp custody, have the court order the child returned back and you could even lose custody if the other side fights it properly

If there is pending litigation and you don't show up a warrant can be issued for your arrest.

No. But she could be defaulted, a judgment entered giving him custody, and she could be ordered to deliver the children. If she doesn't, then, she could be arrested.

Depends on the nature or stage of the court case.