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Dear ninam - All of these private agreements are unenforceable. Your fiance needs to file a motion with the court for an order of visitation since he cannot do anything to enforce whatever he and the mother agreed to. The same applies to all aspects of the children from medical to child support to custody and visitation. without a court order he has no rights to do anything.
If he had a court order at this point he could file a motion for contempt of visitation against the mother and get some action from the court. At this point he has no way to do anything. The issues concerning your father are irrelevant since every case has different facts. The child support issue is based on the respective income of the parties and who has the court ordered custody.
You fiance needs to file a court case to get all the issues resolved since this will be going on until the children are adults.