Retema Park in Selma, Texas has quarter horse racing in June. Unfortunately, Manor Downs seems to be closed.
AQHA: Racing - American Quarter Horse Association is the governing body.
Quarter horse racing is a straight line normally, with no turns. Distance is normally a quarter of a mile.
If I remember right, THE VILLAGE HORSE DOCTOR, by Ben Green, has a story about a quarter horse race in Texas about 1950. That race was run the way they would have done in Natchez.
Starting point in 1819 is the distance would have been whatever straight line they found, and would have been less then a half mile. But they wouldn't have been too hung up on exact distances back then, so the quarter mile would have been approximate.
Horses would line up at one end, more or less in line. Race would normally start upon a gun firing. Lots of jostling and other acts that would now be considered fouls. Race could be anything from two horses to twenty or more. At a certain point, they might divide the race into two or more.
Mares, geldings, and stallions race together. Cultural influences would determine a lot of this. In Texas, most of the horses would be geldings. Stallions were ridden more by Mexicans.
Jockeys would often be kids, frequently Mexican or Indian. But it depended. There would have been no weight requirements, so the lighter the better. Jockeys would use whatever worked. If buzzers had been invented (a device that provides a mild electric shock that is placed under the saddle), they'd have used them.
Heavy betting occurred. Ringers were common. And it was only cheating if you got shot when caught. It was not uncommon to use two horses on either side of your competition to ram into the competition at the start. Anything to win. And judging the winner was sometimes hard on the eyes. The horse you thought won by a nose wasn't actually there. Natchez had a bad reputation in honesty.
Races were a common part of holiday festivities. Often there would be several races, with different criteria for each race. As has been constantly the case, choosing the right race to enter is an important part of winning. (Many people are not aware that a week before the Kentucky Derby is the Kentucky Oaks, limited to fillies. Even a filly that is competitive in the Derby might be entered into the Oaks to make it more likely to win. But that's one reason why fillies are few and far between in the Derby.)
Best of luck,
Jim Clark-Dawe