Why do people join them? What do they do for their members?

House Redoran values strength of arms and rewards strong warriors, which means strong warriors see value in joining the House, on the other hand Redoran tries to keep its people and holdings safe. Being under Redoran protection counts for a lot when you're stuck in the hinterlands of Velothis (and Vvardenfell), which attracts people who want to survive.

In general, but especially in the more wild hinterlands, the Redoran can be quite meritocratic. While they are definitely wary of the local Nords and Orcs, they will work with them and will tend to treat them relatively fairly. While they do not offer a path to riches, they are trustworthy.

At the same time, House Redoran places a lot of importance in family, but not in bloodline. That means that Velothi (and others) can be and are adopted into Redoran families and can reach powerful positions in the House.

This contrasts somewhat with House Hlaalu, which likewise provides opportunities to those from lowly backgrounds, but can be a lot more sink or swim: ambitious members of lowly background can make it really big, or end up dumped in a canal with a dagger in their back.

What factions splintered off of them?

None. They do occasionally split off bandit groups, and technically several Ashlander clans of old were Redoran and proto-Redoran splinters.

Do they have internal factions?

Aside from the natural decentralised state of affairs (the Marches being intra-Redoran blocks of interest), the house is relatively stable. Most Redoran are happy to work within the framework of the House and are not trying to transform it into something completely different or have goals that are at odds with a significant part of the House. A certain amount of stagnancy and complacency in terms of their role and structure is pretty much one of the major issues that faces the House and contributes to its increasing irrelevance. The only recent shakeup is the radical "faction" headed by Bolvyn Venim. who moved the capital to Ald-ruhn and embarked on the hopeless crusade against the Blight, as well as a reactionary counter-"faction".

What are they doing?

How do they sustain themselves?

Foraging, farming (either communal/clan Redoran farms or Velothi yeoman famers under the Houses protection), mercenary works, protection deals for caravans and such. While the House itself is strong about martial prowess, they defend those they rule over and they pay most of the bills.

What is common knowledge about them?

House Redoran is known as boring, honorable, and pious.
Alternatively a noble house of warriors there to protect Morrowind which has been gathering dust, and has been left in the dust (such as when Vvardenfell was opened for colonization and they failed to capitalize on their position quickly enough). At the same time, it is acknowledged that they should not be provoked too much; while they allow themselves to be restricted by the laws of the Tribunal and (now) the Empire, their strength as warriors is still highly estimated.

What do members know about their faction that outsiders don't?

Very little, as Redoran does not really have secrets. They would be aware of the organisation of the Marches and their superiors, and peers, as well as their charges.

What dirty secrets do the in-the-know people know?

The Skar can still walk, if properly motivated.
House Redoran took in a lot of Sixth House members after the War of the First Council, and should have a bit more insight, through half-remembered oral history, about House Dagoth.

What are the three most notable events in faction history?

The Nord Expulsion - proto-Redoran started out as a bunch of war-like clans that constantly fought each other, and were consequently quickly conquered by the Nords. Nerevar uniting the Dunmer as a whole also turned these clans in particular into a cohesive and cooperative fighting force. As the Nords were finally pushed back through Rift Pass, Kogotel was established as a fortress to stop them coming back, with a council hall for the proto-Redoran clans. The surrounding territory was turned into a march. While before House Redoran’s inception which would happen after the War of the First Council, this lay the groundwork for the formation and structure of the future house.

The Four Score War - after a long period of peace where Redoran had little to do and was losing its edge, Morrowind suddenly faced a huge threat in the form of Reman Cyrodiil. The Four Score War was an 80 years war, fought largely in western Morrowind’s Redoran and Hlaalu territory, and was a far more impressive clash between gods and god-like figures and dragons and the like than Townway presents its last year in his 2920 series. While Redoran performed impressively in the war the destruction greatly weakened the house in general and the Waters March in particular.

The Opening of Vvardenfell/Moving the capital to Ald-ruhn - while Redoran had ancient holdings on Vvardenfell, in respect for the island’s status as a Temple reserve (under Imperial protection, as cemented in the Armistice) Redoran chose not to reassert its claims on the island when it was opened to settlement in 3E 414. As a result, House Hlaalu and House Telvanni occupied a lot of territory Redoran felt to be rightfully theirs. As the last in a long line of set-backs and territorial losses Redoran had suffered over the course of its existence, House Redoran’s principled stance proved very unwise, and after becoming Archmaster Bolvyn Venim made the radical decision to move the house capital to Vvardenfell and drag the various march lords with him.

Bolyn Venim moves the council to Ald-ruhn

What happened to them during and after the Armistice?

After Dagoth Ur’s awakening, the resulting ash and blight storms washing over part of Redoran territory, and the losses sustained in the Tribunal's unsuccessful campaign to regain access to the Heart Chamber, House Redoran had little appetite to fight two wars at two fronts at the same time. 

As such, the Armistice was, although begrudgingly, welcomed by the House, as it averted a war Morrowind was ill-prepared for, and solidified Redoran's claims to the Uld Vraech, the Snows’ March and turned Vvardenfell into a Temple preserve under additional Imperial protection.
However, at the same time the Water March became unstable and dissolved in the face of powerful Hlaalu expansionism.

What happened to them during and after the Simulacrum?

The Arnesian War - sparked in southern Morrowind. House Redoran joined the crusade, but while ultimately victorious performed poorly.
House Dres was not entirely cooperative due to hiding the secret origins of the conflict (the Hist deal), the Redoran troops had to fight outside of Morrowind territory in the Arnesian Jungle, which they were not equipped to handle and some of them may have seen some pretty bad stuff there. Indoril and the other Houses were only giving token assistance and contributions and might have to some degree destabilized things, and to some degree tried to steal Redoran's thunder.