Though I would give a shot at answering these, since horse riding is all I did last night when I got the game.
1.) Horse riding is a skill, it goes up the more you ride a horse. This includes the racing competitions and the jumping competitions
(as below in #2) or just doing your own courses at your sims home or park.
2.) There may be opportunities related with the Competitions, I'm not sure, but they are all done in the rabbit hole. Around noon and 5pm sim time, the competitions open up, and you can pick which of the three competitions you want to participate in.
3.) You can buy a horse from the Equestrian Center, make one in CAS, or your sim can try to make friends with a Wild Horse found around town and adopt him into the family.
Just realized I've been reading this site for quite some time and never registered here...
And it looks like I was beaten, by a smidge, but I'll post it anyway.