Perhaps. Sheogorath has allowed a mortal to mantle him, after all. Though it does seem to go against their nature...
I'm not sure. Maybe Proweler, Nalion or another Lore-Khan will have something to post later...
Yes and no. Jyggalag and Sheogorath were two sides of the same coin. One could argue that from a mythic point-of-view what happened was that Jyggalag (who wanted to be free to do what he wanted) split the coin in half and split from Sheogorath. The Champion of Cyrodiil only got in the way and was like a fly who was spending too much time on an open highway.
Okay, maybe the anology is off. Let's try this again.
Jyggalag, buried inside of Sheogorath's subconscious, wanted to not be buried inside of Sheogorath's subconscious. However, in order to do that, Sheogorath would have to die and unfortunately the conscious part of Sheogorath (our friendly and loveable daM doG) didn't want to die. So, what he did was created an out for Sheogorath by having him die and yet not die. Sheogorath did not want to die (which he technically did every Grey March), so he took the out.
Anyways. My uber point is this: Sheogorath, like all Daedra, are survivalists and tyrants. They don't want to share power because it might mean that they would end up becoming undefined (as good as dead). Even in the context of influence over the Mortal realm, they do not want other players. There are 28 players already, so the game is complex enough as it is. Add four more... phaw.