I am installing a ceiling fan. I ran new 12/3 wire for the fan. The existing switch in the wall is powered by 14/2. I want to run the lites off the switch and the fan with constant power. Can I tie the black source wire to my black fan wire & then pigtail that connection to the bottom screw on the switch, and then connect my red fan wire to the top screw on the switch, giving me switched power to the red wire & constant power to the black wire? Also, this is on a 20 amp breaker, is it okay to use 12 gauge wire tying into the existing 14 gauge wire?

If you are using a 20 amp circuit breaker for this circuit, then all of the 14 AWG wiring must be replaced with 12 AWG cable.

14 AWG, cable is rated for no higher than 15 amp. It cannot withstand the heat generated by 20 amps.

You really have two choices, remove all of the 14 AWG conductors from the circuit or replace the 20 amp breaker with a 15 amp breaker.

One or the other only!

Other than the 20 amp breaker you are fine, black to black and pig tail to switch, white to white and red to switch. Does not matter which wire goes to which screw on switch.