All disciplines. Show names are usually longer and more formal/fancy. You don't have to have one even if you are in shows but they just sound nicer sometimes.
They are allowed in Jumpers
Generally show names are longer and more formal. At the shows I go to for example, before entering the ring the judge will announce something like "this is Dances in Moonlight, ridden by (riders name)" but the horse's barn name is Pepper. Hope that made sense:)
A show name is just the horses official name that is on their papers. Like my horse's show name is Infiniti but his barn name such is kind of like a let name is Finn. I believe that there are show names in all disciplines. You can also have a horse who's show name is the same as their barn name.