Do horses know they are racing and do they know whether they have won or not?

Well most horses are competitive, running along side of each other they will jockey for the lead in most cases. They have been trained depending on the horses ability to either jump out in front and stay there, or holdback and go when asked for an opening. Not sure if they know about winning, but they have succeeded in their try.

They have no clue, they run faster if they've been shown that theres carrots and sugar cubes waiting for them to finish. All that other stuff is Hollywood nonsense. Unless your Bob Baffert, he shows them milkshakes 😭

Some horses seem to know what they are supposed to do, some not.  Some horses like to run with other horses.  So if they get in front they slow down and let the others catch up.  The bettors don't like that very much.

partially.  Unfortunately, they also realize they are being hit by sticks by little men on their backs.

In the wild the horse does not have to be faster than the tiger. He merely has to be faster than the slowest horse in the herd. Some horses actually do like racing, and especially in the spring you will see a bunch of younger ones (and occasionally some old ones, too) running back and forth as if it was somehow crucially important. 

Both Secretariat and Seabiscuit seemed to act so.

they might not know any of that

Yes! Horses loved to run and are naturally dominating and very competitive! They also understand the positive energy and excitement around them, or the lack of. It's a great experience to witness. Plus, they all have their own unique personalities too. A horse's spirit and their will to win is an important element in the making of a champion.