Yes, I've always called the leg protections used in jumping and such "horse boots". There's also shoe/boot things that go around the hooves for riding in rough terrain, which I've heard of but never seen in person.
In North America, we also use the term "barn" as a building for horses, though it's a more general term, as other farm animals can also live inside of barns, like cows, sheep, etc.
I got the answer - stable, but I live near Louisville and Lexington, KY and there are a few horses around here. Most everyone that I know in the breeding, racing fields, and people in general call them barns - even at Churchill Downs and Keeneland Race Tracks (both of which have been visited by Queen Elizabeth, several Saudi Princes, and people from all around the world). I think if you know "barn" it isn't too much of a stretch to come up with "stable" if the first one doesn't work. And Diva, most of the stables that I have visited have cats living there, too. So what would you call them?
A mule is the offspring of a male donkey and
a female horse. And a hinny is vise-versa.