Yes, horses have feathers. No, they can't fly with them.
Feather or feathering is long hair that grows from a horse's fetlocks. Light horses generally produce very little feather, which is normally trimmed off for neatness. Most draft horses (some, like the Suffolk, are clean-legged) and some pony breeds grow extensive feather that may cover the feet and extend up the rear of the legs.
This Appaloosa mare is "clean-legged" and in her summer coat. She has almost no feather.
It is not acceptable to trim the feather when showing on draft and pony breeds known for having full feather. In the pony world this is part of showing "au naturel" - that is to say, the pony should be shown looking like the only thing you have done is brush it.