Does Coolant have anything to do with my A/C?

Actually these are two separate systems.  Coolant, or Antifreeze, is the colored fluid (usually green, orange/red,  or blue) found in your radiator. In the summer months it keeps the water in your radiator and engine from boiling over.  In the winter, it keeps it from freezing, hence the dual name.   Your A/C system uses a type of refrigerant (commonly called Freon)  to cool the interior of your vehicle.  Having a coolant flush performed  will not help your A/C stay cool, or visa versa.