Correct. The heat pad (unless malfunctioning) never heats up enough to melt/soften/damage the plastic. At the highest setting, I'd wager a heat pad gets in the low 100s as far as temperature, which is well below what it would take to affect plastic wraps.
No need for a thermometer, just use your hand to make sure it's operating correctly (evenly warm throughout).
As far as the height, at the lowest they should be able to sit while touching/nearly touching the HMP, and at the highest, be able to stand and touch/nearly touch it. If it's too low...
Honestly I don't think heat plates are very accurate with their numbers. A large heat pad (like 12"x24") would be good for maybe 16 chicks until weaned, so I too would question how 20 would fit under a 12"x8". 20 bantams, maybe.
Can't answer on the Thermo mat as I've never tried it.
Your brooder is tiny for 15 - what's the next step in your plan? For reference I keep 3-4 chicks in an approx 2.25 x 4' brooder. You may want to jump to the next size up brooder instead.
As far as a heating pad set up (which is just a...