Do show horses ever get stage fright?

The general answer is: No. Most horses don't seem to actually be stressed by an audience per se, although some horses may become afraid of specific things. For example, I have known more than one horse who was terrified of applause. Some horses are also bothered by flash photography, umbrellas, dogs, and small children. Riders, of course, get stage fright - and the negative performance effect sometimes seen when in the show ring is more likely because the rider is tense and not performing well.