Probably for communication, like wolves.
Yes. I am not aware of any evidence that coyotes (or dogs or wolves) howl at the moon. A brighter moon provides more nocturnal illumination, which allows the animals to become more active. They therefore communicate more. [Other explanations: (1) the moon is their god, and their howls are prayers or invocations; (2) the lone moon inspires feelings of mournful, soulful longing, which they express through piercing, evocative howls; and (3) they want to eat the moon and can't, which makes them cry out in frustration.]