no they did not mongols would not even dare to blink if there general told them otherwise. They simply stayed awake for days until the general had enough.

My uncle said, yea, we use grab a few winks when ever Ol' Khan wasn't looking

Each Mongol horse soldier had three or four horses, and when traveling long distances and one horse got tired, they would change horses without stopping. The horses stayed fresher and they didn't lose time stopping.
The Mongol horses were a smaller breed of horse, they were bred for stamina, not speed like an Arabian horse.
They could close their eyes and get some type of rest, but I'm not sure about getting real sleep in the saddle.

Each soldier having three/four horses had another advantage also. It tricked the enemy, they would see a large dust cloud coming and think it was a force of Mongols three or four times larger than it actually was.

Certainly that is possible. Remember warrior had a string of horses they switched between. Horses can run 10 miles an hour x ten hours(divided by five horses at 2 hours each)=100 miles.

Mongols were trained to endure great hardships, and eat and drink sparingly when necessary. For example, they put raw meat under their saddles at the start of each the day, and ate the tenderized, warmed meat at night.

Yes, all horse soldiers throughout the centuries learned to doze on horseback during forced marches. Not sleeping, but a kind of rest.

The Pony Express riders in the western US did the same speeds and distances