Of course they do. The shoe keeps the horse's hoof off the ground. Please keep your horseshoes ALWAYS on your horse. Some horses hooves grow incorrectly, so shoes protect and correct them. If you are in jumpers, or hunters, your horse should get a shoe that absorbed shock, and otherwise, unless your farrier says so, your horse DOES NOT need shoes. Or, if the horse feels off, them you should talk to a farrier. And horses shouldn't get any cuts or bruises, unless they step on a rock. They should have a lone enough hoof, that the frog is a little off the ground. Also, the wall of the hoof is strong, and if it is gross out, You should get some bell boots.
Sometimes horses don't have to wear shoes because they have a special kind of hoof that can take anything.
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No because most Mustangs are wild. The tame/domesticated mustangs wear shoes only when they have to or are ridden in areas that may damage their hooves. But their hooves are harder and tougher than most, so in most cases shoes are not necessary.