Electrical – Are electrical switches under kitchen sinks common for any utility

A GFCI is a good idea because of the location of the outlet. However, I don't think that a GFCI that can simultaneously protect two circuits even exists (and I doubt one would fit into a single-gang box if it did exist), so I believe you have a couple of options:

1. Install GFCI breakers on the dishwasher and disposal circuits, and use a regular outlet under your sink with the tab between the screw terminals broken off. This may be what you already have; the breakers will have a Test button on them if so. If the supply for the disposal runs to the switch first then to the outlet, this is the only way that the switch will be covered by the GFCI. In your place, this is what I would do: safer and less work, albeit a little more expensive.

2. Use GFCI outlets. As I said above, you'd need two GFCIs to protect your two circuits, so you would have to install a double-gang box in place of the existing gang-box or add another single gang box then run the wiring for one of the appliances to it instead (in your place, I'd do the latter and use a remodeling box for the new outlet). However, this means that the switch for the disposal wouldn't be covered by the GFCI, potentially risking an electric shock.

The receptacle that the disposal is plugged into is fed from the Load side of the GFCI receptacle, thereby providing ground fault protection to the receptacle. There is a ground fault somewhere between the GFCI and the disposal, which is causing the GFCI to do it's job and trip.

You'll have to locate the fault, and fix it. If you don't have (or know how to use) a multimeter, you'll want to contact an Electrician. If you have a multimeter and know how to use it, you can narrow down where the fault is by checking continuity between hot and ground.

Start by turning off the breaker, and insuring power is off in the area you'll be working. Unplug the disposal and do a continuity test between the hot blade (smaller blade) on the plug, and the ground pin (long roundish pin). If the test shows there is not fault in the disposal, move up the circuit to the next device (receptacle) and test again (don't plug the disposal back in). If you still haven't found the fault, continue working your way up the circuit (the switch next) until you reach the GFCI. If you still haven't found the fault, try replacing the GFCI receptacle (making sure to hook it back up the same way).

If you still cannot resolve the issue, it's probably best to call in an Electrician (since they'll have cooler gadgets to troubleshoot the circuit). I would focus most of my attention on the disposal and the switch, since those tend to be the usual culprits.