Why should photons emitted by a hotter gas have, on average, shorter wavelengths than photons emitted by a cooler gas?

According to wein's law,

wavelength has inverse relation with temperature

so that's why hotter gas has shorter wavelength than cooler gas.

hope this answer is correct

a lot of it means it can be "burned" for energy and sold at one price. if it is scarce, the price will go up

it doesn't. it was imagined up in the middle ages.