Could a "non-dimmable" LED lamp be dimmed with PWM?
I suspect that the standard wall dimmer an electrician installs is just a variable resistor. 


Incorrect. That would get very hot and waste power. Dimmers use triacs to adjust the fraction of time that the AC is allowed through.



Figure 1. Phase angle control of AC used in standard incandescent lamp dimmers. As switch-on point is delayed power is reduced.


  ... you can get solid state relays ... Since they are solid state, I suspect I could turn them on and off fast.


You could turn them on once per mains half cycle. They would then remain on until the next zero-cross.


  What happens if I PWM a standard "non-dimmable" LED lamp? 


It will probably fail as the internal power supply is not designed for chopped supply voltage.


  Is there some complicated circuitry (that I don't yet understand or know of) that will interact poorly with a PWM power source?


Yes, quite a bit of both.



For more on this subject see my answers to


Confusion with TRIAC firing and zero crossing point.
Activating SSR for an AC motor via PWM input where much of this is explained.
PWM with alternative load.