How about a half height door with a top part that is always open, so that my horse can look out?
This is not a new idea -
Leaf suggested a half door in 2012:
...and Journeya asked for a stable door in 2015:
The closest current solution I can find is an archway with a gate in it, which is nice, but not very solid, so how about one of these:
Just some ideas.