My question, here, is what requirements must your party have to be able to complete each Might and Magic game?

For purposes of this topic, let's require that the party must be able to complete the game in a reasonable manner; for instance, parties that require a specific random loot drop (like an item to cast a mandatory spell) do not count as being able to complete the game.

As for what I know:

MM2: Party must have at least one non-Robber character, as a party of all Robbers can't complete any of the class specific quests, which is required to beat the game.

MM4: Party must have at least one Sorcerer or Archer thanks to the need for the Teleport spell. (Why can't Druids and Rangers use that spell?)

MM5: Party must have at least one Sorcerer, Archer, Druid, or Ranger in order to cast Jump, which is mandatory.

Any information about the other games in the series? Can they be (reasonably) beaten with arbitrary parties, or does the game expect your party to have specific capabilities? (Can MM3 be beaten without magic, for example?)

Can't you have any items to cast spells like Jump/Teleport in MM 4/5 to avoid the necessity to leep certain characters in party? Or it counts as relying on random loot drop in your book?

MM6+ don't have any special party requirements and can be completed with any setup. But some people who try to compelte them with all Knights, solo and/or at Level 1 are exploiting game engine. If you don't exploit game engine, you will have a lot of problems if you won't have characters who can give you access to Self and Elemental Magic (Cleric and Sorcerer being the classic examples).

MM3 can be completed with any party setup, but some specific areas can't be accessed without Teleport, those areas are optional and aren't needed to complete the game.

The point of my "no relying on random loot drop" rule is to disallow that case, as items that cast such spells only appear in random loot or randomly in store inventory.

(Also, why can't Rangers and Druids learn Teleport? That limitation doesn't fit the pattern for which spells they can use, and that limitation also limits viable party selection unnecessarily.)

I assume you are only talking about characters you create in your party. In that case, your comment on MM2 is not true. All you have to do is go on a single class specific quest with one hireling, and you can get all 6 robbers (or 7 with another robber hireling) their + in one fell swoop!

I have heard from others (take with a grain of salt) that MM6 is not only beatable, but easily beatable (for a repeat playthrough not a first-time playthrough), when playing with a SSSS, DSSS, or CSSS party, but can also be beat with an all might-based party (which is hard but doable), so I believe no class is essential in MM6.

With MM8, you only create one character and can hire up to 4 other party members to join you. There are several options to recruit from. So, again anecdotally (I have not beaten MM8), I don't believe there is an essential class.

Ehhhh...with MM8 lava is a huge pain in the butt. I don't know how you're supposed to beat the game without Levitate (i.e. Vampire)?

It is entirely and easily doable. Just don't linger on the lava and you'll be fine. There aren't any areas where you can't at least dash across it from island to island, and in most areas there is some land route from objective to objective. The fire based prison might be the only area in the game where you have no choice but to run across it a few times, and even then you can try to walk on wall edges to help mitigate the damage while running from bit-of-land to bit-of-land.