Here you go tempest for LCD https://github.com/luamfb/tempest-lcd

Kinda both.

FM is normally transmitted at much higher frequencies (~ 100 Mhz rather than ~400 kHz). Your monitor pixel data will have components up at 100Mhz, but they'll be much smaller.

The math to generate an AM signal is fairly trivial (just multiply signal by carrier and re-quantize to the pixel clock). For FM, the math is much harder, although still doable.

I think it is a hardware limitation--the monitor operates at a fixed frequency, and adjustments to the image only change the amplitude of the signal.

Re: FM

Probably because the frequency range would need to be in MHz rather than KHz.