Yes horses can and do lay down when they are comfortable with their surroundings and have the ability to. Horses can also lock their back legs to sleep standing.
Yes horses lay down but they do not sleep while laying down they stand up. But they can sleep lunging and they get on their back and roll around but dont worry they get right back up.
Yes they do. Horses will lay down and rest but they actually don't sleep like that. They sleep standing up because they were hunted by other animals for years.