Does the MOON give off any heat?
Yes, it does, technically. According to thermal radiation theory (first link), any mass not at absolute zero will eventually radiate energy away from its surface. This radiation is a form of heat (the others being convection and conduction). Since the moon is a mass, a large one at that, and is not at absolute zero temperature, it must emit thermal radiation. The second link shows a white paper discussing the topic in detail and even given the radiation’s physical power: the exceedingly low energy wavelength of roughly 1.25cm. This is so low, in fact, that it often times is simply ignored.