Stamina issues aren't really important, rounds in the game last less than 10 minutes.
Cavalry's ridiculously slow on foot, since if they weren't they would be like foot officers on steroids - cavalry rankers have double 1h WPF than officers (as you might have noticed how easy dismounted cavalry can kill) and armored cavalry can tank hits and kill enemies while they're stunned (also they can survive hits which would kill any other class). They're quite powerful on foot as it is, making them faster would just unbalance them.
Light cavalry is only good against unprepared opponents and lancers. Good infantry can forever stay out of the light cavs tiny sword, while good heavy cavalry will slice you to pieces if you even try to close in. However, their fast horses allow them to pick battles they can win.
Heavy cavalry are unmatched in sheer killing potential, their only weakness is getting shot or getting their horse poked to death by banking lancers (which takes quite some time). In melee, infantry can't stand against heavy cav on open ground, light cavalry will get killed before they can even reach them, while dragoons, if they don't get lucky driveby hit, will get slaughtered even easier. Also, dismounted armored cav is extremely deadly when clearing out the houses (and similar places where running speed isn't so important).
Cav vs cav balance is in good state (not counting dragoons here, they just get slaughtered by everything) as is cav vs inf (again, not counting dragoons, as they just get slaughtered here too).