I'm in a somewhat similar boat to the OP. I have 10 acres in southeast Nebraska. We just built a barn, and we're about to bring home our first horse this weekend (although I grew up with horses).
As for mats, a horse friend recently turned me on to StallSavers.com
. He uses them, so I've seen them in action. They're great. They're only about 1/4" thick, but they're porous, so liquids run right through to whatever's underneath. They're made to size when you order them. They're not heavy, so they have to be secured around the edges of the stall--that may be a problem for your configuration. I'm just about to lay the floor in my stalls, and I plan to put down about 2" of crushed limestone over top of the moderate clay that's already there. I'll put Stall Savers on top of the limestone.
As for the tractor, QtrBel is spot on. Figure out what your toughest job is, and get a tractor that will match that job. If you need to look up specs of a tractor, use TractorData.com. My acreage came with an old Ford 8N, which is handy for many things, but sucks for horses. It has no front end loader, and the rear 3-point lift is too wimpy. I've been shopping for something more powerful, but not so large that it's not maneuverable. I like the Kubota L3??? series (or similar from other brands), which is about a 35 hp tractor, 4wd, with enough lift capacity to handle a 1500-lb round hay bale. Recent models have a quick-detach loader that's compatible with skid loader attachments, which sounds useful.
A 40hp tractor is usually a jump up to a larger chassis, which may or may not be harder to maneuver around your property or inside your horse pens. Bigger is not always better.
Well-maintained, used tractors hold their value really well. If you buy used & take care of it, you can probably sell it in a few years for what you paid if you decide you want something different. A used Kubota L3??? will run about $14-18K around here. I've never priced them new.