If your horse falls in the river and gets stuck in the water just choose another horse and you'll get back on land.

You need at least a tier two or a tier three horse to complete the first relic race, don't even bother if you don't have one. Use stamina boosts

In order to visit a friends paddock go to the trophy icon at the very top of the screen, in it on the top left you'll find the leaderboard and should tap it. If you're connected to Facebook and they play the game there will be people in that leaderboard, tap on the blue "I" and then tap "visit" to see their paddock.

You can rename your horse by going into the inventory in the paddock.