As far as the windows go, changing the shape of the window would require finding windows that fit in the current hole and then re-skinning the hole with round corners.
Simply put it would be a major PITA and you would be much better off putting original type square windows back in.
Or you may consider just having a upholstery shop make you some canvas snap covers that can go over the openings you can throw on if it is raining or cold to keep your horse out of the elements and outside of that not worry about it which is what I would do.
With a trailer the age of yours i'd be much more worried about the condition of the floor, if it is wood and it appears dry rotted at all replace it. The last thing you want is your horse falling through the floor at 70mph, this has happened before and it is not pretty when it does. Especially if you don't notice it for an hour or two.
Other things I strongly recommend are replacing the lights with LED's and heat shrinking all of the connections as well as using SO cord to run all the wires.
You should also get your brake system in proper order including the break-away switch and battery.
On your axles you should also replace the caps with bearing buddies, they allow you to add grease to your axles with ease. I have never had a bearing issue with a trailer running them.
Also I like this little puppy here for your break away cable. Fastway Zip Coiled Trailer Breakaway Cable - 6' Long Fastway Accessories and Parts FA80-01-2160
When you get tires, stay away from Carlisle's or the other cheap china brands, they tend to blow out fast. I run either Good Year or Michelin, preferably Michelin, but they are also more expensive, but they also last forever and ever and ever and ever.
With the age of the trailer also inspect the wheels, you want to check that they are not bent, and that there is no rust forming around the lug nuts or the periphery of the rim. What can happen is the rust can cause the lug nuts to not seat properly, or in extreme cases pull through, and on many of the formed steel wheels they can rust out around the welds and crack there.
As for paint, I always paint my trailers with rust-oleum it is great because you can buy it in 1 gallon cans, then touch it up with rattle cans that are available everywhere under the sun. If you paint it when it is above 70° and not very humid the paint will dry nice and hard and last for a good long time.
Oh and another must do thing before you go towing your horses in it is thoroughly inspect the condition of the coupler and safety chains, if there is any doubt as to their condition have them replaced by a reputable welder. Sure it will cost a few hundred bucks, but that will be cheap in comparison if you were to lose the trailer, especially if it happened on the interstate cut loose and hit someone else'es vehicle.
Also you will want to inspect the condition of the tongue jack, if it operates fine id just recommend adding some grease to it which can be done just by smearing it on, the jack leg towards the top when fully extended.
Other items you will want to have, wheel chocks, a foot or lumber for the tongue jack to rest on.
I do always always always chock a trailer when I un hitch it from my tow vehicle, and I will also chock it if on any kind of an incline when parked, and on that note when towing and you park somewhere, set your parking brake, and make sure before you tow that brake is in good working order.
If you have any questions feel free to bounce them at me, I have over a million miles of towing experience in 47 states, 3 Canadian provinces and Mexico, and I used to own a welding shop and did a lot of repairs on trailers as well as building new ones.
Good luck with your new trailer.