This is a criminal law, rather than a child custody law, question.

What will the court need for proof of adultery and marital rape?

Rape should be reported to the police immediately.

How hard is it for father to get sole custody ?

Sole custody is difficult to obtain when there is suspicion of possible future harm to the child, but no real evidence to support the suspicion.

Is my custody agreement still valid?

An attorney would need to review what you are calling your custody and relocation "agreements" to determine whether they are still valid....

Can my ex-husband keep my kids from visiting my residence in another city if it’s not stipulated in the legal paperwork?

A judgment signed by the judge in your case (not just "the legal paperwork") is what controls who can do what regarding the kids. If one parent...

Can a mother show up after being in and out for two years and hasn't seen him in four months take from father

Yes the mother can show up and take her child unless there is a court order or judgment prohibiting it. The father can file for custody, though, to...

Wrong dad on birth certificate and he has custody of child I need answers/advice (not one that says contact a family lawyer)

I understand your desire to resolve this via a Q&A on the internet, but the harsh reality is that, if you are hoping to establish paternity, you...

Can child support service change an existing support order implemented in family court ?what can they do if order exist already?

Child support orders are not final while the relevant child(ren) are still minors. Either party (or the State) can seek a modification.

Options for change of custody across state lines?

You can hire a Louisiana attorney to file to change custody in Louisiana and to obtain a court order addressing your concerns.

It's mystifying (though sadly common) as to why he has allowed two years to elapse without seeing his children or filing for custody. He needs to...