I don't think we do add "from" to any of those verbs.
Some people add an unnecessary of, as in I fell off of my horse; I got off of the bus; I took the cover off of the dish. It isn't considered good style, but you may hear native speakers using it. There are some previous threads on that usage.
The word "off" already conveys the idea of movement away from the horse.
off (WR dictionary): in a direction that is away from a place: to look off toward the west.
There's therefore no need for "from".
And similarly you could just say "fell from my horse" without "off", as #2 says.