Person with breathing problems wants and needs very cool air. If system is kept at about 65 to 70 degrees in warm and hot weather, could this low setting be causing freeze-up? All parts of the AC except ducts have been replaced within the last five years. System is big enough for the house. Freeze-up continues. Advise please.

65 would cause freeze up. Anything below 72 is hard to do. Also dirty air filter or low on refrigerant and dirty coils will also freeze up. I would call in a qualified tech in to do complete service yearly. That way unit will always be operating at max and you will always have the cleanist air circulating throughout the home.