My husband's ex literally handed over custody of his two kids to him. She just "didnt want to do it anymore". Well they agreed that she would have them every other weekend and that she would pay child support. Can she at any point and time fight and get them back? They sent the final paperwork to the courts and its final. He NEVER wants to give them back to her besides what they agreed on. But won't the state look at her and think "a mother should never give up her children" Doesn't this though process help him and hurt her?

Get the paperwork finalized and keep documenting everything she does that could be considered neglect or abuse. If she were to try and get them back the court would see that your husband has the primary custody and he is what is the best interest of the children so she really wouldn't have much of a chance to 'win' at anything.

...Can she at any point and time fight and get them back?...
Modification of child custody can be filed most any time because the law and courts want to protect children and do what is in their best interests.