1. Is there any distance where a reasonably fit human can regularly outrun a horse.

2. or a horse carrying a rider.

Reminds me of the Cristiano Ronaldo vs Bugatti Veyron challenge

Well, a horse can gallop at around 30 miles per hour, whereas an average human being can run at around 17 or 18 miles per hour. So really, there's a vast difference in speed, but I honestly do not know if a human being can actually outrun a horse in a given distance.

About the speed of the horse's gallop, and the speed of an average adult when running are both rough estimations from the results I got on Google. So yeah

What if they were to race 100km?

A horse can gallop for only 3 miles, trot for 15 miles.

Well, I'm not sure about that at all, but I'm sure it depends on the terrain, as well as the fitness of the horse. Otherwise, I don't know

Flat terrain. Reasonably fit horse. Running on dirt or grass. We try to keep things as even as possible.

I think there was a program (maybe on Discovery Channel) where they pit a man against a horse or some other animal. The man lost.