Does the mat produce FIR rays when not plugged in? If so, how does it compare to when it is plugged in?

Same as with ions, the crystals do not stop generating FIR rays.

When the mat is unplugged, they use the heat of your own body. 

The body heats up the crystals and when warm they emit rays in so called “crystal” wavelengths between 4 and 15 nm. 

Amethyst concentrates most of heating energy on the two peaks, at 7 and 12 microns.  

Of course, the more warmth crystal receive, the more they emit. 

So, when you heat up your mat under sun (like in ancient times) or by electric system, it gets really hot and emits more energy. The wavelength is the same. 

It does not mean that you should try to use the mat always on the highest heat. Unpowered or low temperature ranges are much safer and more gentle.