The normal answer is, whatever you really want to happen to them. The Nerevarine was only rumored to go there by NPC's who otherwise know nothing else. The same goes for the other every other hero in the series.
Is Nerevar immortal like the Tribunal?
Not quite. The Tribunal depended on the heart for their powers, while the Nerevarine is now in a sort of stasis, effectively rendering him ageless and immune to disease. He can still meet a violent end however.
Where is the Tribunal now and what do they do?
For all intents and purposes, they're dead and they're busy being dead. You can make the case that Vivec is still alive, but the fall of Lie Rock means hes effectively not around for the moment.
Is it possible for the CoC, Nerevar, and the last Dragonborn to be together all at once
If you're CoC wasn't the one who mantled Sheogorath and is a Mer, there's no reason why they couldn't get together at some point. Its whatever you really want to happen.