How should I address my bathroom's flickering lights and GFCI tripping mysteriously? (Schematic included.)
Sure sounds like you may have a couple of unrelated problems. Usually, a nuisance tripping GFI is a sign of a worn or aged GFI device. This is assuming there are no down line devices or lights that are leaking small amounts of current to earth ground. This would be uncommon for a light or simple switch, more likely with a motor load on start up or shut down.  The more important and potentially dangerous problem is the whole house lighting flickering when the fan and shower light is on. If you have an accurate digital VOM, you can look at the voltage drop at the breaker that controls the bathroom. If you see variations of much more than 5 to 10 volts then isolate which device is causing the fluctuation. (fan or bath lite) by turning them on individually and test again. A older fan is a common source of your problem as moisture and humidity may have invaded the motor and causing larger than average voltage drop to ground. If this fan is in fact the culprit, the fan may be heating up more than normal as well and could become a hazard. If you do not have a VOM or know how to test voltages inside your circuit breaker panel, I'd advise you to seek a licensed electrician soon to correct the situation.