It's more movement based, maybe taking a trainer or someone who is familier with the hunter world when you go to try horses would be a good reference?
Paints are good for anything but thoroughbred have an unfavorable build for flat classes. (In my area atleast they are less then desired)
breed doesn't matter as much as experience. as long as they are safe and can teach you.
thoroughbreds are really versatile. i have always had some type of warmblood, they are great w jumping and most look great while doing it (hunters).
I have a paint and A quarter horse and there both amazing