Yes they do , it's part of their nature....always ready to flee from prediators.
Whenever I see horse down sleeping in their stall, which is rare, my mind sometimes panics and I think colic or something. Nice looking horse you have there too!!
Horses have a 'stay apparatus' in their legs that they 'lock' when they go to sleep standing up. Sometimes you can hear a very soft click in an older horse when they unlock it.
Ours can choose to be out or in. Their stalls are their 'mane caves'. They generally lay down in their stalls, unless it's absolutely perfect outside - warm, sunny, a light breeze, and not a bug in sight. Then it looks like our pasture is full of dead horses...stretched out flat, LOL. My neighbor's horses are the same, further, he's got a mare and foal who will NOT stay outside, they sneak in and lie down in the stalls during the day, LOL. But they have an ulterior motive - they prefer to stay in the shade and nibble up the little stray bits of hay laying around.
Them laying down in their stalls doesn't necessarily mean colic. I look for changes in their chosen routine. Such as them lying down in places they don't routinely choose - and in unusual positions. A fine mist of sweat on the foreface is also a tip off, as is a distant, preoccupied look. They may look dull or depressed. Or - look back at their belly, try to pee and get only a few drips, etc.
I found that rolling over is indicative of colic sometimes. Unless you have a horse that just likes to roll a lot. Usually you can tell if it's because of pain though. We had a mare with colic one time. Some hay was too stemmy and she got it. It was our sons horse so he got to walk it around with a lead for 3 hours till the vet could get there. We didn't have a trailer at the time.
Horses do like to roll. The difference in a normal roll and a colic roll is, generally, when a horse rises from a normal roll, it will shake off. A colicky horse will appear distressed and look for another spot to roll again and again. I have ten horses, all of them LOVE to roll.,