OK. Checked with my wife (12+ years of dressage riding, Animal Science degree, 7+ years in a vet's office; but I'm recalling this from a midnight conversation last night, so my recollections may be imperfect) and lots of horses wear booties.
Miniature horses are sometimes used as therapy animals in hospitals -- they wear booties to cut down on the clip-clop clip-clop.
Horses can get an infection or inflamation of the hoof called "thrush"; a bootie filled with ointment is fastened around the hoof to soak it.
Horses that "overstep" wear "bells" around their hooves (but not under) to keep them from stepping on themselves.
The horses don't need any special training or conditioning to get used to the bootie -- ties around their ankles don't bother them alot, and whatever's underneath is just another surface to walk on. Unlike dogs (who also wear booties on occasion), horses don't have toes or nerve endings in the bottoms of their feet (hooves are dead material, just like fingernails), so they're really not so aware of the bootie.
On a related matter, horses wear horseshoes if they walk on hard surfaces alot, or they don't have a pasture or large area to run in. Horses that are let out to pasture usually have their shoes pulled, while one that works in a city will be shod (the shoe actually acts as a buffer, spreading out the impact of the stone on the hoof and protecting it). Nowadays shoes are also used correctively -- to keep a horse's hoof from spreading (or splitting?), or even (in theory) if one leg was slightly shorter than the others, to even things out with a "high shoe" (this is my example, not hers).
A horse might also be shod if it works on slippery or steep surfaces -- the shoes can be studded, giving the horse much better traction.
Mules and donkeys can be shoed -- most of them just don't need to be nowadays.
Hope that helps,
Nell.