Since no one has answered your question about horsie behavior yet, I will.
Horses generally don't sit like dogs except when getting up from a laying position and will not lay down unless they feel relaxed and comfortable with their environment or they are very sick. In other words it's normal for them to lay down in their pasture or stall, somewhere they feel safe, but if he's laying down in your trailer it may be from some kind of colic or stress shock. You could recognize that by the look of distress in the horse's eyes, possible sweating on their chest and body area, and elevated pulse and respiration. A resting horse will look calm and happy, often have their ears forward or relaxed to the side, and may rest their nose on the ground by their forelegs. They will lay flat out for short periods of time or roll to scratch their backs but usually do not stay in those positions for long.
I'm glad you're asking questions, please feel free to ask away! It's nice to see a new owner researching so diligently.
Leia