If the custodial parent agrees with non-custodial parent to allow children to reside with the non-custodial parent, then has the custodial parent voluntary reliquished custody. Also does the custodial parent pay child support to the non-custodial parent?

Simple answer is no to both. You would have to go back to court to have custody switched and support paid to the new custodial parent. If it is just an agreement between the two of you nothing changes concerning the legality of the case.

Was anything ever signed at court relinquishing rights to the non custodial or was that just a verbal agreement??
If it has gone through court then yes the non custodial would then have to go through court to get child support that they would win since the child IS with them.