The atmosphere is spinning with the Earth. It has to be, otherwise you'd have 1000 mph winds at Earth's surface. How? It gets dragged along by friction with the surface
I'm not a physicist but it seems to me that given entropy and the thinning of the atmosphere as elevation increases that clouds would not a) not appear to float in place and b) only move in one direction.
Low clouds would appear to float in place but high clouds would move quickly away from the spin of the earth.
Not sure how someone would go about proving this with household tools
It's reflecting sunlight right, so why does moonlight register as colder than shaded spots.
It's a real easy and fun experiment.
One guess is that the Moon is not a perfect reflector and the photon density per unit area coming off the Moon is a lot lower.
You could test this hypothesis by pointing a laser thermometer at a light bulb and a pile of dirt reflecting the light from the bulb. The Moon is basically made out of gray dirt.
It would be interesting to try the experiment on a moonless night to check the difference between open and shaded ground. If they are the same the shade doesn't likely matter (given a sufficient time from sundown).