Should engine run cooler when AC is engaged?

If you have electric fans that are controlled by the A/c it will run cooler. Otherwise not usually. If you have a thermostaticlly controlled clutch fan it may engage shortly after the a/c is turned on. The a/c condenser (by design) purges heat from the system and because it is in front of the radiator, it usually heats the cooling system up. A/C cars have larger radiators, larger fans and usually shrouds to aid in the cooling.