I've seen wells that fully replenish the mana of a hero that visits, but since that is just a map feature, I can't always rely on it to be there. How else can I restore the mana of my caster heroes?
You can also restore mana by spending 1 turn in a castle/town that you own that has a mage guild built in it. It will restore the entire mana bar.
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There are eight different short audio clips played at the beginning of every battle in HoMM III. The corresponding file names are battle0{0..7}.wav (contained in one of the .snd archives). How does the game choose which one to play: is it random, or does it depend on the battle terrain / other factors?
It's completely random from what I remember.
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In Heroes of Might and Magic III your heroes gain skills as they level up. When you level up, you will be presented with two options for skills, one of which you can opt to learn. There are three tiers of each skill.
Another way to gain skills is to get them from random objects in on the map.
Because I don't look at what I'm doing and click anything that says unvisited, I always end up ruining my hero by learning a skill like "Scouting" or "Eagle Eye" from a random dwelling.
Is there a way I can discard these skills?
No, there's no way of unlearning skills in Heroes III. At least not in the core version of the game, WoG has introduced the Arcane Towers that allow a hero to redistribute his or her primary skill points but not secondary skill points. (Wake of Gods)
A good tactic that I use is to visit Witch Huts and similar dwellings with my secondary hero to avoid messing up my primary heros's skills.
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