Fall damage is calculated by calculating the distance fallen (in blocks).
Specifically, Falling Damage =(number of blocks fallen x ½) - 1½.
This means that you can safely drop 3 blocks at once without taking damage. Jumping gives 1½ blocks worth of upward momentum, so if you jump off another block, you can only fall one and a half blocks safely.
Ladders, water, and other blocks can mitigate this. Ladders can by cancelling your momentum (and resetting your fall distance) and water because it's water, and generally stops falling damage in any video game that fails to model surface tension!
Horse armor doesn't have durability, it'll only be removed from the horse if you equip different armor on the horse. As such it is not repairable. I do not know if it can be enchanted however.
Best Answer
No, armour does not get damaged from falling. This is tested in this video.