Don't install a mini split if the environment is more dusty than a typical home. The evaporators and squirrel cages clog with dust very readily and are a pain to clean.

If the AC is not going to see a lot of use, meaning not a low set point or not operating to keep the room cool for many hours, you won't see much savings over a typical window unit which are becoming more efficient all the time. They are very inexpensive and if they irrevocably clog up, you can throw it out an put in a new one.

Do you want heat gain on the West side? If you are in a hot clime, put a wide roof overhang on that side

If there's a roof opening above feed room-stall corner,any of the options can be mounted in that corner. Otherwise the nearest low point in shed roof need's to be higher than outdoor coil. Ionized make's a good point about M splits getting dirter and harder to clean than others. I overcame that issue with auxiliary filtering of indoor air before it enter's unit. Basicly it's a 1/4tr" plywood box on wall below unit(could be sheet metal). Needless to say,the hvac tech isn't going to install box and how the box look's is dependent on woodworking/sheet metal skill one has. In your tack room it doesn't have to look great and should be easy. There are no plan drawings nor specs. Each case must make sure fan has more cfm than evap fan. Up to this point,this is well in reach of the average diy gearhead. The final tweak is adjusting static pressure with an instrument called manometer. Many techs don't have one and a diy certainly shouldn't buy one. If this interest's you and you are confident you can build it but tech doesn't have the meter,want's more than $50 or refuse's,ask and we can get it close enough by trail and error.

The mini split does not have to be on an outside wall. It is much easier to install them that way. If you are willing to deal with a condensate lift pump, or otherwise find a way to drain it, you can put it anywhere. If there is a drain in the tack room, feed room or in the main aisle, you can use it.

Thank you for your advice. Is the evaporator the indoor unit? I'm pretty sure that inside the tack room would be more somewhat more dusty than a typical home, and I'm sure that outside would be quite a bit more dusty, especially where the green outdoor box is drawn, since about 30 feet away to the west is the riding ring which has stone dust footing, and my wife drags the ring every couple of weeks.

The open lean-to area has a roof over the entire area. At the location of the label "Roof line" the height is about 8 or 9 feet. There is a drain line installed in the concrete almost exactly beneath the location of the yellow indoor box.

As far as putting a wide roof over the west side, the building and roof are already built, so its probably too late.

I don't seem to have a good close-up picture of the barn, but I found this one taken from the next hill over.

You have it. "Evaporator" refers to the inside heat exchanger in a dedicated cooling device commonly called "air conditioning" or "air conditioner". The pressurized liquid evaporates inside it cooling it. Outside is commonly called the condenser because the hot, compressed gas coolant is cooled turning it into a hot liquid then a cool liquid. If we are talking a heat pump, the positions are reversed in heating mode so there is sometimes confusion.

I would not worry about an outside unit with respect to dust. You might need to put a regular cleaning routine in place that the average homeowner does not, especially if it is under cover where rain does not reach it. I don't really have the experience to say one way or another, but the cleaning should be relatively easy.

Given my personal experience, I would not put a mini split indoor unit in a dusty environment unless the installer would say that it is not a problem and refund your money or install something else if you find it clogging up.

I am sure that the YouTube mini split cleaning vids have proliferated since I took up the problem and I have not reviewed the situation for a year or three. They should be worth a look, but beware that if some look much easier than others, they may not be doing a complete job. My feeling is that it is best to keep up with the buildup and it may prevent the need to do more complete dismantling, but I don't know for sure.