My fist thought was to mix a bunch of cement this spring and lay that down, creating dry ground. Someone said, however, that I would also have to put re-bar braces in so the cement doesn't crack under the miniature horse's weight.
First I think you have a good little building to start with. Seems solid enough and easy to add onto eventually also.
Ok I'll address your damp floor problem and the cement. Yes to make a concrete floor properly you will need to build your forms (we usually use scrap 2x4's) and rebar and pieces of left over field fencing. You'd need to pour it about 3 to 4" thick I'd imagine if that is the route you want to take.
If not, I would get a truckload of 1" gravel and pour it super thick. When we did our barn floors, we put in a lot more than needed because then we ran it over continuously with 4 wheelers and the tractors to pound it down......then added more. On top we were very generous with sand. Part of my stalls are matted and part of them are not. The ones that do not have mats, are the biggest pain in the rear and are the cause of many of my grey hairs and my loss of jovial lollipops and rainbow sense of humor. Seems like when you dig out your pee holes, they get deeper and deeper each day and it never stops because most of my horses have that certain designated pee place. Watch for mats to go on sale at Tractor Supply.
Also I'm looking at your tin roof. Be sure when the ground starts to thaw, that the roof is not "raining" from condensation. I'll bet it is causing a lot of the dampness inside. I had that happen in another barn and it was the death of me. Soaked my horses and stalls all the time and ruined my bedding constantly until my barn became totally combustible so watch out for that one. Its a biggie. That's when we went to tar and shingles on our new barn.
Best wishes to this very exciting endeavor.
This shows the inside of the stalls. I placed a board where they eat because they love to paw during dinner time. Drives me batty.
This is the back view of the stairs up to the loft and porch deck where I like to sit.