Can you run 14 gauge wire off 12 gauge 20 amp breaker?

You could if it fit properly in the connector, but it would be massive overkill since you only need 12 AWG wire. You also have to make sure the load can physically accommodate that larger size wire.

Every so often someone will replace an electric cooktop witha gas one. All of the new gas cooktops require a 110v outlet. The old electric cooktop may have been run with a 50 amp breaker (and probably 6 gauge wire) In order to convert this 220 v circuit to a 110v circuit we change to a 20amp breaker. Depending upon the brand of breaker sometimes the wire will fit into it. If not, we splice on a short piece of 12 gauge wire and then connect to the breaker.