My tack room sounds like it has a similar floorplan. Same size, with a window straight across from the door. You will quickly find that 12x12 doesn’t give you a lot of “floor space” along the walls.
In warmer months, I use a temp controlled window fan, or a window AC unit. The room is super well built, draft free and heavily insulated, so I am able to keep it “comfortable” all year round.
My barn cats live in my tack room, too, so I have a covered litter box and their food dishes in there.
I have an old/antique formica topped metal kitchen cabinet and an antique tall locker in there. They are priceless for dust-free rodent free storage. Even though my tack room is well sealed and draft free, dust always finds a way in!
Saddles are hung on the wall, to save floor space. My tack room has a 11’ ceiling, so the less used saddles go on racks up high that require a step stool to reach. The saddles in constant use are lower, but still about 5’ off the ground. Having them that high allows me to have a HUGE patio/deck box below them to store sheets and winter blankets in, and allow the lid to open fully, and me to not knock my head when I am getting stuff from the box.
Bridles are hung on racks tucked between the saddles, and on the wall on each side of the window.
I have 3 Rubbermaid trash cans in the tack room, as it also doubles as my feed room. The length of wall above them is lined with hooks, spaced about 6" apart, that have all manner of halters, leads, breastplates, cruppers, martigales, etc. hanging on them. They are up high, so nothing hangs down in the way of the feed cans.
I have a very well supported shelf that holds three 60# vittles vaults with dog food, cat food, and ration balancer in them. It was custom built so that those cheap plastic 3-drawer units fit flush under it, with just enough room to slide in, but they bear zero weight so as not to crush them. I have bits, leg wraps, quilts, first aid stuff, etc. stored in the drawers.
The remaining floor space along the wall is where I park the Dyson. Yes, I have a Dyson solely to vacuum the tack room! I also squeeze the space heater in there, and a couple re-purposed 50# CortaFlx tubs being used for more storage.
Having everything along the walls gives me a generous floor space in the middle to safely park the space heater in the winter, or a saddle stand for cleaning saddles, or a Momma goat with newborns that need warmed up, or a zero gravity recliner for me to sleep in when I have to monitor somebody overnight in the barn.