What are the restrictions on spellcasting for "might" heroes in Heroes 3 again? For example, Barbarians and Knights can't cast spells above level 3 or something, right? Or can they cast any level of spell, but the mage guild in "might" castles (Barbarian, Knight) only goes up to level 3 or something? What is the rule?

Then, there are also "might" heroes for magic castles too like Gunnar for the Warlock castle right? Does he have spellcasting restrictions too then?

Here's the thing:

No hero without the Wisdom secondary skill can cast spells above level 2. Magic heroes come with Basic Wisdom by default, which is why they can when you hire them, although nothing stops Might heroes from acquiring the Wisdom skill. Advanced Wisdom enables a hero to cast level 4 spells, and Expert Wisdom enables level 5 spells.

Mage Guild maximum levels differ from town to town:

Level 3 - Fortress, Stronghold

Level 4 - Castle

Level 5 - Rampart, Tower (+ Library), Inferno, Necropolis, Dungeon, Conflux

In Heroes III, there are no strictly Might/Magic-oriented towns, and all have both Might and Magic heroes. The only exception to this rule is the Might town in Heroes IV, which doesn't have Magic heroes of any description nor Mage Guilds, but that's another story.

In general, any hero can learn any spell if they meet the requirements. A hero must have the Wisdom skill to lean any spells above 2nd level, with each level of wisdom granting access to another level of spells (so Expert Wisdom is required to learn 5th level spells). A hero can cast any spell they know as long as they have enough spell points (with a few spells having specific extra restrictions, like you can't summon a boat if you aren't next to water).

Some castles can build level 5 Mage guilds, but other castles have a lower limit. Particular maps may have more restrictions than these; these are just the general rule.
5th: Dungeon, Tower, Rampart, Inferno, and Conflux. Edit: and Necropolis.
4th: Castle (the human town)
3rd: Fortress, Stronghold

Each faction has a branch of Might oriented heroes and a branch of magic oriented heroes. The specifics of how those heoroes tend to develop varies according to their branch, with even the might heroes of the Tower being more magic oriented than the magic heroes of the Stronghold. Not all magic heroes start with wisdom. Stronghold is a prime example of this.

Um.... They actually do. http://www.heroesofmightandmagic.com/heroes3/heroesstronghold.shtml

Hm. I could have sworn there were some exceptions, but maybe I just never payed close enough attention to the specific class. When it was time to buy another hero, I picked the specific (available) one that better suited my needs.

I'll try to take a look in-game in a few minutes.

I was right, just really wrong about the faction: http://www.heroesofmightandmagic.com/heroes3/heroesnecropolis.shtml

The Necromancer faction has Death Knights (Might) and Necromancers (Magic). A lot of the necromancers do not have Wisdom. All of the faction heroes have the Necromancy skill, though.

Looks like every magic hero for every other faction starts with Wisdom, though.

Ah, that sounds right. Necro is quite unique in more ways than one.

their main disadvantage with spellcasting is their lower mana pool and power.also certain might hero types cannot learn some magic schools except from a witch hut:
barbarians and overlords cannot learn water magic secondary skill,rangers and beastmasters cannot learn fire
everyone can learn earth and air ,magic heroes also can learn any from these 4 magic schools

It is worth noting that these heroes can still learn all of the spells (assuming high enough wisdom), they just can't really specialize in the relevant schools of magic.

Wouldn't visiting a Magic University be an exception to this?

Two more ways to learn forbidden skills are scholars and map events. As said before in universities and Conflux magical universities all forbidden skills will be on red backgroung and impossible to buy.

And there are more forbidden skills for certain hero types:
- Resistance for Warlocks, Witches and Wizards;
- First Aid for Death Knights and Necromancers;
- Leadership for Death Knights and Necromancers;
- Navigation for Battle Mages;
- Necromancy for all heroes! (non-Necropolis heroes can't learn this skill, all Death Knights and Necromancers already have it).

I'm pretty sure at least part of this list is incorrect; I'm pretty sure I've passed on acquiring the Necromancy skill for non-necromancers.

It's not possible in unmodded game, unless in one of beforementioned ways (scholar, map event or witch's hut edited in map editor, because default settings exclude Necromancy and Diplomacy from skills available on hut's list).

What about Knights (the might hero from "Castle" faction)? Are they restricted from learning any of the magic skills (air magic, water magic, fire magic, earth magic)?