Looks like a Shoop trailer. I had one very similar, but older. Solid trailers.
That floor wood looks good from here. They are commonly made of oak, so if they aren't split or rotten, they're fine. Take an ice pick, or an awl, or even a small phillips screwdriver and poke around in the corners and where the wood meets metal. If you can't stick it into the wood easily and don't see anything obviously rotten, it's sound. Don't replace it. Replacing the floor wood is a major project and 2" oak is not cheap. For sure, don't replace it with treated lumber. Treated lumber is made from a variety of pine or fir, and it isn't nearly as strong as oak. Not only that, but the chemical it is treated with reacts with metal and speeds up the rust process where ever it touches metal.
Those floor mats you have look good to me as well. I wouldn't buy new ones. Floor mats are amazingly expensive for a piece of rubber.
Dark color on the sides won't increase heat too much, but have them paint the roof white if you can. That will make as much as 20 degrees difference inside in the summer. Horses inside a dark trailer in the summer, even with windows fully open, can overheat very quickly, even if you just get into slow traffic on the highway. Overheating a horse is serious business and can be fatal.
Install two good vents in the roof.
For saddle racks, you'll need to have some cross braces welded to the upright braces in the front of the trailer to attach the foldable racks to. The metal skin simply won't hold it. Or you can lay a piece of 1/2 or 3/4" plywood on the floor, screw it solidly to the floor boards, bolt a metal upright to the floor and mount your saddle stands to it. You may have to make a place on the tongue for your spare tire, since I believe Shoop trailers normally mounted their spare inside. You might also consider looking online for a tack box or water tank with built-in saddle racks on top. I've seen those and one would work very well in your trailer without modifications.
Rather than worry about welding in a feed shelf, which will cause you problems with the doors, you might consider buying hay bags. That's what I use. Just stuff them with hay and hang them from an eyelet in an upper corner.
When restoring and remodeling an old trailer, keep your costs in mind, because you won't get it back, no matter how good it looks, and you will quickly find you have spent more than you would have, had you bought the trailer you wanted in the first place. First and foremost
, get someone to re-pack, or at least check, all the wheel bearings and brakes. Don't take the dealer's word for it, unless you saw them do it or specifically paid them to do it. It doesn't take long and you will have the peace of KNOWING the last time it was done. Trailer bearings and brakes very seldom get checked until they go bad. Getting caught out on the road late on a Saturday evening with a burned bearing is a real bummer.