To minimize the damage to an over heated pan (be it with or without food), do I let it cool off? Run it under cool water? Or is there different approach I'm supposed to take?
Letting it cool slowly will minimize the chance of warping for thin or cheap cookware.
Quality cookware should stand up to being deglazed or quickly cooled under running water.
How you cool it is up to you.
Note: the above is based on metal pots and pans. Glass, ceramic, or earthenware pans should always be cooled slowly to prevent heat-stress breaking or crazing.