Is the money taken based on a customary agreement after a quarrel lawful, and is it permissible for my father to dispose of it as he wishes?

If someone assaults your father, and the assault results in financial harm or damage, your father is permitted to take the value of the damage or to reconcile with the aggressor. If what these individuals ruled for your father is equivalent to the damage, then there is no impediment to it. However, if they exceeded that without the consent of the one against whom the ruling was made, then your father has no right to it, and he is obligated to return it to the one against whom it was ruled.