Electrical – 20amp 12/2 wire with 14/2 switch

You are correct that the red should be power to the light, except the other side of the switch should take off from the line in side of the GFCI, there is no requirement for lighting to be protected by GFCI, so do not connect to the load side. The GFCI tripped because there was a mismatch in the currents between power wire and the neutral wire attached to the GFCI.

This is based on my best guess based on the information you provided. To be certain, you should examine the wiring in the other switch box to ensure the black is unswitched power, white is neutral, and the red is connected to another wire leading to the light. The main uncertainty is how the other switch plays into this scenario, as there were not enough wires for a 3 way setup with power to an outlet as well.

Incidentally, I would suggest avoiding the push-in terminals if binding screw connections are available. Doing so is more work, but it provides a better more reliable connection, provided the binding screw is properly used.

You should re-feed this fan from a lighting circuit or other 15A circuit.

A ceiling fan CANNOT be supplied from a kitchen or DR receptacle circuit. Whoever did this was clueless with regard to codes and proper wiring.