During a battle, you can cast many spells, and each will reach its target. Mana is irrelevant here.
...but why cast ten spells at once? Usually, only two or three are used at most. A single-target spell, an area-of-effect spell... you can use a different element. No one invests in counterspells, and they don't always work – it's much better to cast something yourself than to waste an entire turn on it, so you don't need to worry about that.

With a Warrior hero, you can destroy half of the mage's army on the first turn.

How? With a Knight? A Dragon? Half the army? I highly doubt it. Only the Furies in the Dungeon have high initiative; the other units aren't known for it. Besides, a mage's army can be comparable in speed and initiative to even a demonic army, especially when equipped with mini-artifacts and/or buffed with Light, which the Academy has no problem with.

You can also try this strategy with the Undead; their troops aren't particularly well-protected or fast either.

It's a complete failure; the Dungeon's unique troops, even when fully buffed, won't be able to do much besides using Subdue and Berserk.

A hero with only one magic skill isn't a mage, and not even a combat mage; it's a warrior with magic...

The most common build is a Warlock with only Chaos, and I wouldn't call him a magic warrior, you know ;D
Very few people invest in two magic skills – mainly, only the Academy.