Some surge protectors work by "redirecting" surges to the equipment ground, which is pretty much the definition of a ground fault. In this case, it's definitely possible that surge suppressors could be tripping GFCI receptacles/breakers (given a large enough surge).
As for the AFCIs... When electrical contacts separate, they often create an arc between the contacts. This arc spans the gap until the contacts are too far apart, and the arc can no longer be sustained. It's possible (in theory) that the GFCI tripping could create a large enough arc, for long enough to cause the AFCI to trip. It's not likely that a properly functioning GFCI would do this, but if the GFCI was old and worn it could be opening the contacts slow enough that a detectable arc is being produced.
Determining if the GFCI tripping is causing the AFCI to trip, is a simple test. Simply press the test button on the GFCI. If the AFCI trips, you'll have figured out half the problem. If the ACFI does not trip, you could repeat the test using a receptacle tester with a GFCI test function.
If it's determined that the GFCI is causing the AFCI breaker to trip, replacing the GFCI receptacle may resolve that part of the issue.
Figuring out if the surge suppressor is tripping the GFCI, is a more difficult proposition. It will likely require you to monitor the supply voltage over some time, to identify and document possible surges.
From the description, it sounds like the neutral may be shared between the two circuits, which you can't do with a GFCI. The GFCI is looking for an imbalance of current on the hot and neutral lines, so if the neutral is shared with another circuit, there will be some current returning on the neutral that did not go out on the hot.
If you can trace the circuit and find the connection between the neutrals on the two circuits, and isolate them, that would be the easiest and cheapest fix. If the neutral is shared via a multiwire branch circuit from the breaker and can't be split, you can change the gfci breaker to a standard breaker and install a gfci outlet after the neutral splits for the two branches. And I'm also seeing two pole gfci breakers which would allow you to replace the two breakers with one that provides gfci coverage to both circuits and should handle the shared neutral, which may be a good fall back option if you can't locate the shared neutral.
Best Answer
Try a Lamp Debuzzing Coil on the line-side of the dimmer
This sounds suspiciously like your dimmer is generating some sort of switching trash on the power line that's interfering with the GFCI's circuitry and causing it to trip, since it only happens if you change speeds. The general fix for this sort of interference problem is to wire a Lamp Debuzzing Coil (LDC) (which is a prepackaged choke, basically) in on the line side of the dimmer, as documented in Lutron appnote #519.
If it persists, change the GFCI out for a new one of good quality
If the Lamp Debuzzing Coil doesn't fix it, then I would keep the coil in place (it'll cut down on RF trash coming out of the dimmer in any case) and replace the GFCI with a new one of good make and quality (you want a specification or industrial grade GFCI receptacle or deadfront device here, not the cheapest builder-grade thing you can find).