Do they actually... you know, do anything except pointlessly get sent to their deaths by the caliphate?
If you're not Jewish? Then no, they're as useless to you as one of the backwater tribes in Siberia (no offense to said tribes in Siberia).
If you're Jewish? Well, they're not the head of their religion per se, but they do get a bit more interesting as Gothic Emperor said once the Kingdom of Israel is created. When Israel is created, they'll eventually "migrate" or make aliyah to Israel (i.e. become your vassal). Being of the House of David, they might end up getting weak claims on the Kingdom of Israel, particularly if you decided not to claim descent from the House of David yourself. That means there's always a chance there will be a claimant faction war that wants to knock you off the throne of Israel.
Also, there's a few special events - somewhat more on the maintainence side - that ensures they'll have healthy male heirs (since, well, mercenary bands usually odn't pump out kids), and also destroys the exilarchy if it is no longer held by someone descended from David (i.e. some random courtier inherits it).
Are there any restrictions for their appereance? I actually never cared for the exilarchs before you mentioned them in this post, but when I searched for them I only found the Karaite Exilarch creeping arround.
The exilarchies appear in most starts except for some of the late ones (like after the Mongol invasions or something). They'll continue to exist so long as the title is held by someone with the "House of David" trait.
One's the head of the Karaite faith IIRC. The fun really starts when someone creates the Kingdom of Israel; the exilarchs get claims on that title.
They aren't head of the Jewish faiths per se, I think there was a bit of discussion ages ago as to whether the exilarchs should have been the head of the Jewish faiths, but for various reasons we decided to make them faux mercenary companies as we felt it was a better way to simulate the historical reality.