Horse Archers combine the strength of archers which is an effective long range attack, and the strength of cavalry which is high mobility, but don't really inherit either of their weaknesses.
The current balance problem isn't so much that cav or archers can't handle them, since it's typically easier for a grounded archer to win a shootout (better archery proficiency and less movement resulting in better accuracy) and cavalry outspeed horse archers through both the riding skill and the ability to buy faster horses.
The gargantuan balance problem is that there's literally nothing an infantry can do against a horse archer. Infantry can easily beat archers by getting in their faces and using their melee advantage, they can easily beat cavalry by using polearms and stopping horses that eventually have to get close enough to deal damage.
Horse archers on the other hand don't run into either of these problems. They don't ever have to get close to deal damage to infantry with their highly mobile horse, so there's no way for an infantry to get close to a horse archer.
Your suggestions wouldn't change that, they'd just make fights against grounded archers nearly impossible and shooting down cavalry much harder, all while making the class extremely unrealistic, resulting in very unnatural and lame horse archery gameplay.
It would be better to just leave horse archery out of multiplayer entirely. If implemented realistically, it would be too powerful. If implemented with the necessary nerfs, it would be disgustingly unnatural and provide both **** gameplay and a bad competitive spectacle.
There's no good reason to have a class that can't be naturally balanced which becomes unenjoyable to play against, or a class that can be artificially balanced, but becomes odd and unenjoyable to play as.