Correct to add the snubbers to both the triacs and the gate driver. the trigger voltages are coming from the mains wiring having some intrinsic inductance, capacitance and this causing a voltage drop on triggering, causing the other ones to switch on ( remember dVdt is only rate of change, and turning on the one triac causes a pretty big pulse as the inductance of the wiring in the supply makes this voltage drop), so you do need the snubbers. Adding some overvoltage protection as well is also a good idea, and for the fan motor a series inductor to reduce the current spikes as well. The worst loads to switch with a triac are pure inductance and pure capacitance, as the turn off is very difficult without snubbing, and using high dVdt triacs as well.
As to the OP, the fan motor does not use a capacitor to control speed, it uses a tapped transformer, to switch in what basically is a lower supply voltage as you want to drop speed, so the motor slip increases and thus the lower speed. Better fans are wired differently, the speed changes are done by having multiple poles, and the windings are arranged so that you change the number of poles of the motor. high speed might be 4 poles, medium 6 poles and low 8 poles, all using the same rotor and stator windings, just having the windings of the coils spread differently on there.
Adds to cost a lot, as you need to have room for all 3 sets of windings, and a compromise on the start winding for efficiency, but the slip is constant, the losses are low ( because of the large amount of iron in the core) in the core and copper losses dominate. However most are wound with a constant pole pitch, or only 2 pitches, so there are more losses, but there is less copper wire ( or in cheap ones copper coated aluminium now) and iron core needed, and it will be running a lot hotter.