I would guess it's likely sufficient shelter. The first winter we had our horses (2009-2010), we had 3 blizzards. During the actual storms, we put them in their stalls (2 per 12x6 stalls) with both of the Dutch doors closed. It made us feel better knowing they had this level of shelter, but I know others feel that what you currently have is sufficient as well, as long as they can get out out of the weather and especially any blowing wind.
Here's a pic of the one building we had at the time (a converted run-in) taken during one of those blizzards. We now have a second one in similar proportions that faces this one:
Here's how we often found three of the four that winter, even without the stall doors closed:
We were using hay that was too course for them (we were so "green" at this whole horse thing, let alone miniature horse thing) that they wasted roughly 1/2 of every bale and it simply became bedding.
As to whether they like snow? Just like any other animal, some do, some don't...must couldn't give a flying fig either way. It's just another day to them!
Case in point:
This was back in the day when we allowed the dogs in with the horses. Too much horse manure eating not to mention the safety concerns (none of them were too bright when it came to understanding that they needed to stay out of the horses' way) led us to reroute their in-ground fencing system so they can't get into the horse area.
ETA: We now use an alfalfa/grass mix hay for everyone. I have taken the three younger mares off of any grain (since the spring) but will be giving them a supplement feed (all supplement, no calories) starting this weekend just for my piece of mind. Our hard keeper is the only one with consistent grain feeding and she's on Purina's miniature feed along with a rice bran pellet.