Open Path
I'm going to make this short because it's evident by your tone you and I won't agree on Morrowind.
Most of the things you think Morrowind does good, I disagree with. The only exception is the worldbuilding. Most of the things you say Morrowind is unparalleled at, I agree with. But being unparalleled at something doesn't mean you are good at it. It's just that not many games bothered to have the same mechanics as Morrowind: not even ones from the same series. The only thing that is brilliant at times is the main quest. Most of the time the writing is simply serviceable, it does what it is needed but never adds the spark of life to NPCs, which is why so many of them have no personality at all.
Morrowind is not a combat-centric game. It's also not a story-heavy game. What's left are the quest, the exploration and freedom, which is basically what Morrowind focuses on. I don't think any of these are actually good; I can definitely say Morrowind gives the player a huge degree of freedom to use its different mechanics, but that would be it. Unless you are that interested in experimenting with mechanics in a game you otherwise find boring, there's not much to praise here.
So in what branch of your absurd classification fits exactly the two religious faction
They are faction quests, just like all other guilds and great houses. Not much science behind this.
and what approach "should" have every quest of these factions?
To me, a faction's quests should be directly related to a faction. In that regard, Morrowind's quests are fine. If there are faction requirements, I expect those to tie in directly into the faction's quests. Else they become generic, and expose faction quests for what they really are: quests where the "faction" element is simply that you are doing these quests for a particular group.
Redoran faction quests, for example, shouldn't limit you to finish them by the use of melee combat... but simply make the skills favored by the guild useful in most quest. With faction checks, the player must improve the favored skills (by the faction, of course) to advance and have access to new quests and rewards, so makes sense that you use at least some of these skills regularly.
This would make sense if it wasn't so easy to level skills in Morrowind in different ways. Ideally you pick a build and stick with it, i.e. I decide to play a fighter and naturally the factions of choice will be Redoran, Fighter's Guild, or Imperial Legion. But no. I believe you can join any faction from the start.
As I said isn't a specially good combat, lacks balance and a bit of challenge, but fatigue influence, weapon deterioration, unique enemies, enemies extremely diverse use of magic, possibility to fail attacks or spells, attacks types and damage dependence of these types in different weapons, one of the most diverse and dynamic magic combats ever -even if overpowered-, attribute-skill heavy dependent combat, difference in weapons reach, underwater combat, very good looting after victory (good by diversity and uniqueness, even if sometimes overpowering) and some other features that I possibly forgot all makes Morrowind combat a decent experience for a rpg fan, specially comparing with the two simple combat games that you mention in your message. Most rpg ever made have far more simple, action based, less challenging, less rewarding or even less balanced combats, so how can be Morrowind's "one of the worst"?
Because it boils down to dicerolls in a game where the action-part allows you to completely break it. There are no rounds. There are no turns. You attack as fast as you please and then are able to spam potions in your inventory at will. Those are genuine problems with the combat. You asked about New Vegas: I don't think New Vegas has good combat, but at least it is proper action combat in an action-RPG.
Don't get me wrong: you make a lot of good points. Those are all genuine strong points with Morrowind's combat. But we cannot ignore the fact the combat is based on dicerolls, in a game with a first person perspective, and it is about as unengaging as it could hope to be.
An how can be quests that can be solved in several ways
I disagree, many Morrowind quests don't have multiple solutions. A lot of times it boils down to "do the quest or don't do it".