Electrical – wiring a switch mounted next to a gfci

Unfortunately, you can't. For a GFCI outlet to protect other devices they have to come after the GFCI outlet in the circuit. The power (hot and neutral wires) have to enter the outlet on the terminals labeled "line," and leave the outlet on the terminals labeled "load." This way the GFCI outlet can measure the power running to your light on the black wire and returning from your light on the white wire, which is the whole purpose of the GFCI. If those two measurements aren't the same, it turns off. Putting a jumper (what you're calling a "pigtail") between the Line and Load terminals on the device end would effectively bypass the GFCI functions inside the outlet altogether. Connecting the neutral wire coming from your breaker box in the same wire nut with the neutral feeding your light AND feeding the GFCI device will give the electrons leaving your light a direct path back to your breaker box, meaning that they never flow back through the GFCI. It will trip and refuse to reset as long as the light is on, assuming it works at all.

If you want that light to be GFCI protected, you will have to re-run the wire from your breaker box so that it enters the box for the GFCI first. Alternatively, you could splice a second cable to the line in your light box and run it over to the device and then use the black and white from the existing 3 conductor wire to take the power back from the device to the light (with the black wire connected to one pole of the switch and a black jumper from the gold "load" terminal on the outlet to the other terminal on the switch). Either way, you have to run additional wiring. It just depends whether it's easier to add a new cable between the device and the light or to run a new cable from the breaker box to the device.

Put another way, if you want to protect things "downstream" using a GFCI outlet, you have to have at least 4 conductors present in the box, and you only have 3.

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.