Leave the movement controle as it is, it's fun and really good once you're used to it. Also, if horses can take damage, there should be a limit where they won't let you ride, i.e. drops you off, yet it should be impossible to kill them, in my opinion. There should be horse armour, if the ganis allow, and maybe even jousting and such. Being able to use bows atop of horses at reduced speed and accuracy would be another nice addition.
And, if you hit your horse, you deserve it liking you less
