In Fresno (CA) a custodial father of five had to repay welfare for money the mother received illegally.
if on welfare, yes
"reimbursed" ... Not clear whether you mean collects welfare or collects child support, but in either case the answer is, the State will not reimburse the NCP.
Yes, depending on state rules and income limits for welfare. Alimony may not be the best choice for the custodial parent.
Welfare
Usually the case will be closed and the custodial parent will have to re-file in the state where she/he resides.
The custodial parent is legally free to relocate. This is because the noncustodial parent's argument as to why the relocation should not happen has not be presented to the court in time for the hearing and therefore the court will grant the custodial parent's application to relocate unopposed.
He needs to file a modification
He/she can lose custody all together. The non-custodial parent needs to file a motion for contempt of a court order. If the custodial parent continues to violate the order they could eventually lose custody.
Leaves them permanently or in violation of court ordered custody/visitation time? In either circumstances, the non-custodial parent may file an action for contempt of court against the custodial parent and/or file for custody/visitation modification based on the same.
for what?
Yes! Why would you want to keep that from the "custodial parent" anyway?