About the army speed, if we wish to stick with realism, when it comes to ancient/medieval times it were baggage trains that slowed the armies down the most. No army could march faster than the slowest baggage trains. Reducing the amount of baggage trains speeds up the march considerably, becouse the more trans you have, the more problems it create - carts can get stuck in mud, axles can brake etc. This is why romans forced their soldiers to carry a lot of the stuff on their back.
Having that in mind it's rather obvious that an army that contains large amounts of horsmen and horses will move faster, simply because they can use less baggage trains and keep a lot of the equipment carried by horses. This is one of the reasons why mongol and other nomad armies were able to move so fast and why cavalry speed bonus makes a lot of sense.
Ironically, when it comes to forests and snow, horse-centric armies will have the same advantage in these environments, precisely because of the need of less baggage trains. Carts in forests will mean that whenever a tree trunk blocks the road (which might happen quite often in old forsests after the storm) it will have to be removed. And wheeled carts are real pain in the butt to move in the winter as the marching amry turns the road into mud in which thewheels will slide,get stuck etc. So, speaking about realism, khuzaits should actually have a speed bonus in forests and snow, but I would never give it to them in game, as they are enaugh of a pain in the butt already