Do horses like to run?

Mostly - although it depends on the horse and its breeding. I've certainly met plenty of horses who would be happiest moseying around and only run if they think they're under threat. Draft horses tend to be particularly laid back (although there's always that horse that didn't read the breed manual).

Other horses, well, it's a challenge to get them to slow down. Thoroughbreds and other racehorses tend to have a strong desire to not just run fast, but to run faster than everyone else.

Many horses will run in their pasture just for the sake of it.