Are horses in Zelda in an area the same for all players or are they randomly generated? If it's random, do horses around the area change their stats when you reload a save?
This IGN guide on horses does a pretty good job explaining how horse stats are generated. In essence, it's random (aside from a handful of unique horses that have set stats, such as Epona). You also won't be able to know an individual horse's stats until you register them, though.
The stats of the horse may improve if you venture into dangerous areas - but you'll never know how good your horse is until you register them. Their stats can be as low as one star or as high as four in any category.
A good indicator of how good a horse's stats will be is their coat color. From the above guide:
Wild horses come in a variety of colors, such as three different tones of brown, tan, black, grey and blue. While one-colored horses tend to be the better ones (higher stats), those with spots normally have lower stats, with an exception of the tan and one of the brown coated horses, of which those without spots have lower stats as well.
To answer whether reloading a save will regenerate the horses in an area, it appears that, yes, this is the case. From this Reddit thread:
The two spots I've found with these horses are around the giant horse, and on top of a plateau between the canyon with the forgotten temple, and the horse stable east of it. You can save before going after a horse, and reload to change their colors. Once you have one with the color you want (the best horses have pure solid color. Either pure black, pure brown, or pure blue is best, with any color mane), go after it and soothe it.