Okay, good questions.
I'm not allergic to sigs in a thread, although turning them off won't harm anyone. Thought I'd answer that one just in case
For your starting army you've a choice between the five types I mentioned. Bascially, a heavy cavalry army means you'll have relatively many horse-armed warriors with plate armour. These are the chevaliers or cavaliers the medieval french army used, the knights Wallace skewered on his pikes in Braveheart, the Paladins in Age of Empires 2.
For light cavalry, think Mongols. Leather or hide armour, with lighter weapons like spears and shortbows.
Heavy infantry armies don't really excist at the moment, regular infantry armies do, and include a mix of lightly armoured pikemen and decently armoured shortswordsmen (good swords are rare).
This is just the shape of the army you'll have at the beginning. A cavalry army will have infantry to back it up, just like an infantry army will have light cavalry to scout the terrain. It's just a matter of where you place the emphasis.
And yes, light cavalry will be better at hit and run attacks whereas heavy cavalry has the protection to last longer under fire, even if they are not as maneuverable.
Once the game starts you'll have the option to hire more units, train more units and expand the realm allowing you to train its occupants and add those to the force as well. There'll be a random factor there, but you should be able to shape the army you started out with into a force you're completely happy with. By 'advising the ruler' not to spend coin on archers but invest instead in breeding horses and training riders, for example, you can turn your force into a fully mounted one over time.
What army you pick at the beginning does not really matter, although I wanted to allow for some choice at least. Just pick the one you think will work best for you, and you will find out if that is so soon enough.
I'll add details on how to prepare for battle and how to fight when required, no point in boring people with them now
On the depth, I decided the accent should be on the warring. Making alliances, gathering new troops, attacking other players and so on. Which is why you play the general rather than the king.
You could perhaps persuade him (or her if you prefer a queen) to spend less on military, but unless your neighbours do the same that'll just make you a better target for invasion.
As DM, I'll play all the kings myself so I can make sure there's plenty happening for everyone. Kings give you orders (objectives), share their concerns (tell you what's going on in the realm and what your options are) and are always open to 'advice'(your choices will always go). You are basically the unofficial ruler on all things military. The monarch will take care of all the other details.
A brief and later more detailed overview of the armies, nations etc. will show up
here as soon as the site is up. I apologise in advance for any popups you may get.