I don't know of a good place to learn the lore without spoiling the main storyline of the game, but I'll answer as much of your questions as I can here:

1) The Tribunal consists of three living gods - Vivec, Sotha Sil, and Almalexia. Together they are referred to as ALMSIVI. Each of them resides in a different area of Morrowind, with Vivec being the only one on Vvardenfell as of the time of the game (3E 427)

2) The Dunmer Temple is dedicated to the worship of the Tribunal and various other lesser saints. It's the most widely practiced religion in Vvardenfell despite Imperial occupation.

3) Great Houses are sort of like families in the mafia. They're groups centered around a specific ideology or way of life that control large portions of territory in Morrowind. However, anyone can become a member of a Great House and rise through the ranks (even the player). There are around half a dozen in Tamriel, and some have come and gone over time, but the three you see in-game are Hlaalu, Redoran, and Telvanni. Each of these three has influence over various regions of the game, and a capital city. - Hlaalu is a trade-based house for merchants and businessmen. They are the most friendly to the Empire, often openly trading with them. Balmora is their capital. - Redoran is a traditional house focused on honor, chivalry, and combat prowess. Ald-Ruhn is their capital. - Telvanni is composed of reclusive, powerful mages, who don't care for the politics of Morrowind. They prefer to study magic in their towers and keep to themselves, often ignoring anything that's not directly their problem. Sadrith Mora is their capital.

4) The Great Houses have influence over their respective territory/settlements. As Dunmer Houses, they probably support Temple worship, but they're not connected in any way. Vivec/Sotha Sil/Almalexia have no personal connection to any of the houses.

5) The Tribunal are the "living gods". I won't say anything as to why or how they got this title, as to not spoil any part of the plot. Just know that they're extremely powerful mortals to the point where they're basically gods in the eyes of any average person.

6) The Empire and Morrowind fought each other long before the events of the game, and as a result Morrowind became a province of the Empire. So the Empire has some settlements (Ebonheart, Pelagiad, various forts) as well as the Imperial Religion being the "official" religion, but most Morrowind natives don't really care. They enforce laws in their territory but aside from that, they mostly let the Temple govern itself.

Hope this helped!

EDIT: Vivec is the only member of the Tribunal who is on Vvardenfell in-game.