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.
The answer to your last question is "Yes", it is perfectly fine to use a 15 amp plug strip on a 20 amp circuit like you have. That is why the slots on the receptacle go both horizontal and vertical. I'd suggest a plug strip with it's own 15 amp resetable breaker built in. That will protect your lower amp tools.
The second issue would be to build your own multiple outlet strip with a two gang handy box and two 20 amp duplex receptacles and a length of 12 AWG or 10 AWG SO or SVT cable terminated with a 20 amp plug.
Best Answer
You should check if these outlets are attached to the wiring via back stab connections.
Back stab connections can show intermittent connection that could show up when the current demand of the larger appliance is loading the outlet.
If you have back stab connections the recommended course of action is to remove the outlet and replace with a quality one that uses the screws on the side to secure the wires, either by bending the wire the correct direction under the screw head or using the type where the screw tightens a wire clamp behind the screw head. Try to avoid the cheapest outlets that come of 0.49USD in packs of 10 and get one that costs a few USD and is labeled as a better quality unit.