If you not done so get and read Cherry Hill’s book Horsekeeping on Small Acreage. It’s very well done and very comprehensive.
If you put doors on a run-in shelter you’ve converted it from a run-in to a barn. I’d not do that. If you’re providing sound basics let the horse choose where it is most comfortable.* Horses don’t like “caves” so making your shelter a bit larger would be a good idea. Orient the opening so that it faces away from the prevailing winds in the worst weather conditions you face. I’m in the TN Valley. Our worst winds come from the southwest, as they can be strong (35-45 mph) and can carry heavy precipitation before a cold front passes. Then the winds shift to the northwest. They are then quite cold but almost always dry. “Almost” means we can get a lot of “wrap around” after a winter low pressure center passes but it will be snow, not rain or ice. This means I orient my shelters so that the opening is to the east.**
Between about +15 degrees F and +60 degrees F the healthy horse has a neutral heat budget, making all they need as long as they have hay.*** Below and above we husbandry-men must help out the horse with wind breaks, shade, or other material items. Some horses need more than others. This means we follow the Golden Rule of Equine Husbandry: We give the horse what it needs, when it needs it, and appropriate quantity and quality. Remember it’s the horse’s actual need, not our need to feel good, that is the key!!!
You asked about cold weather watering. Google “solar powered livestock winter feeders” and you’ll find a number of examples.
Horsekeeping your part of the world can be challenging but is doable. Good luck in your project.
G.
*It’s intensely frustrating to look outside and see all the horses standing around in cold rain when the run-in is empty.
**Our weather here can be difficult, with LOTS of rain at any time of the year except late summer and early fall. We also get periodic easterly flows and means wind and rain from the east. Fortunately wind velocities from the east are generally low, as is precipitation amount. So the optimum, not perfect, orientation is straight east for us.
***The very act of eating hay for the horse generates heat. That’s why it’s better to increase hay, not supplements like grain, in cold weather.