[Disclaimer: wall of text of doom! Had a busy weekend, going to reply to the entire thread at once
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First, a note on counterattacks: free counterattacks are obviously a bad idea, if a knight is getting ganged up on by 8 peasants at once, he just doesn't have time to simultaneously hit all 8 back. Subtracting movement from your next turn per counterattack is also a bad idea; what if my dragon wants to just ignore all those stupid peasants poking his toes for minimal damage, charging past them to crush your sovereign? You shouldn't be able to deny the dragon movement by sending ineffectual attacks at it; this removes the other player's choices. There's an elegant solution: you can only counterattack if you have spare movement leftover from last turn, and counterattacks use up that movement. So if my swordsman holds his ground with 2 movement saved, preparing himself to be attacked, he gets a couple counterattacks if he gets hit during my opponent's turn; if he charges forward using all his movement and perhaps attacking someone, he's leaving himself open to being attacked himself without the ability to counterattack, which makes a lot of sense. Also if some system is used where different weapons have different movement costs, so you might not have enough moves left to use your weapon again if it's a slow one, this provides some usefulness to those 'leftover' moves (you get to counterattack at least, counterattacks would be cheaper than regular attacks, say half cost).
Also, on a semi-related note, attack speed and movement speed really do need to be separated, as many have already said. How does a dagger make you move faster than a guy with no weapon? Knights with lances shouldn't get to attack quickly, but they need to move quickly; the guy in plate with the greatsword needs to move very slowly, but he should be able to attack fairly quickly (faster than a knight with a lance, at any rate).
Easiest way to implement this, I think, is the suggestion someone made to give all weapons (and spells too, why not) a combat speed stat that represents attacks per turn, such that movement/combat speed = cost to attack - but that's overcomplicating things. Instead of displaying the "combat speed" on the item, show players the end result - the "cost to attack" - because that's the number players actually need to know anyway, and skipping to that part saves them an unnecessary calculation (this has some other unintended effects, which the math-inclined may notice, more on that later*). Say you have 3 movement, and your greatsword has a delay ("cost to attack") of 2. This just means it costs 2 movement each time you swing it; one swing leaves you with 1/3 movement. This leads to an odd situation where you don't have enough movement leftover to swing again, however, which is a problem - less of a problem if that movement can be used to counterattack, but still. It'd be nice if you can swing again at half damage, perhaps; like sweeping a greatsword right-to-left, and following it up with a weaker left-to-right backswing (there's a reason they sharpened both sides of the blade).
*This also lets you create weapons that require a high movement unit to use at all; you could have a lance that costs 4 movement to use, so that even a cavalry can only use it once per turn, and an infantry couldn't swing it at all.. this makes a lot of sense, weapons like that need high movement to build up the momentum to use them.
The main thing we are looking for here is that we want a user to be able to have god like heroes (think Sauron at the beginning of LOTR) and the best way to do that is to make sure that unit can go around and kill a lot of units via either Melee or magically in a single tactical turn. We can't do this if we use moves as the number of tiles they can move otherwise we'd have to let that unit move a ridiculous number of tiles on the main map.
You can make "Sauron" units either by using high stats, or by using special abilities.
Here's how a special abillity Sauron might look:
Sweep - The attack hits three squares in front of him
Reach - The unit may melee attack one square forward
Cleaving - Any time you kill a unit, you make one additional single-target attack against the nearest foe
If they stack, it means Sauron will be able to swing his mace and hit six squares in front of him. If he kills anyone, he gets a free auto-attack against anyone still standing. Boom! Six units go flying backward.
Was going to say this myself, but TCores beat me to it. Sauron does not attack quickly, and neither does a dragon breathing fire at a group of soldiers, or any other megaunit that can sweep through dozens of peasants at a time. These are slow moving, slow attacking units, the key is they can hit multiple enemies at once. Giving them them 5 or 10 attacks a turn so they can knock out 5 or 10 peasants a turn is absurd, because then they can use all those attacks on a single target as well - can you imagine Sauron walking up to a single enemy and whacking it 5x a turn with his huge mace, getting in more attacks than a dagger wielder? They need to do aoe damage, not fast attack speed, and this can be represented by special abilities as above.
Incidentally this makes countering such units much more interesting, merely giving them high combat speed lets them be equally effective against anything, while aoe makes them particularly effective against clusters of weak units - but gives you the possibility of countering them with a single strong unit and/or spread out ranged units (it even makes sense, you don't slay the dragon with a cluster of peasant spearmen it can flatten all at once, you rain arrows on it from afar or send a knight out to melee it).
By the way, abilities need to be linkable to items, so that equipping your sovereign with a claymore lets him cleave/sweep through two or three enemies at once - there needs to be a distinction between light weapons that can attack quickly and huge weapons that attack slowly but hit multiple enemies. Critters that don't need weapons (i.e. dragons) could still have cleave as an innate ability, but no Sauron-type humanoid should have it by default regardless of weapon (can you imagine Sauron sweeping through 5 guys with his bare fists? He needs that huge mace to do it). If you do include some massive Sauron's-mace-like weapon that can cleave through many many units at once, you could add a strength requirement to use it, so that no ordinary peasant can pick it up and start sweeping through armies; it'd be reserved for the powerful, high level units that should have that kind of army-slaying potential, like a properly upgraded sovereign.