The "cute moth" is a white underwing. I added a photo of one with its wings open to reveal its underwing. It belongs to a huge group of underwings, most of which have much more dramatic colorings with red, orange or yellow in various patterns, so the white underwing is quite plain in comparison.
The snowshoe hare eating charcoal is apparently not unusual so that isn't surprising. They and many other animals, wild and domestic, are known to eat some charcoal as an aid to their digestive systems. It apparently helps detoxify their systems which is helpful because some of the things they eat for food, or perhaps accidentally/not on purpose/out of desperation, contain substances that are somewhat toxic to them. Animals seem to know (are attracted to) "cures" for some of the things that ail them and charcoal is one such "cure". We humans use charcoal for purification purposes in a variety of ways. In modern times the science behind what charcoal helps purify is known and understood, but it's likely that our far distant ancestors learned through experience that charcoal had healing properties even if they couldn't explain it scientifically.
Carl Karasti