1. Is there any relation between Jygallag and Hermeaus Mora, both have librarires which contain infinite prophecies and such, as the caretaker said.
Daedra lords often have spheres that overlap. Pyrite's is the "Ordering of the Lowest Orders of Oblivion" or something like that--you think that would overlap with Jygallag's, but it apparently doesn't.
2. The Shivering Isles, is that the name of the land or is it the name of the land when Sheogorath rules it.
Probably the later. "Shivering Isles" is a pretty insane name, since there's only one island and it doesn't shiver. I'd imagine that if Jygallag succesfully took over, the name would be changed to something more logical.
3. Where do the people in SI come from, not mazken or aureal.
The impression I got is that they came from Tamriel, and ventured into Sheogorath's realm when they went insane. I don't think there's any official explanation, though.
4. We have supposedly broken the cycle, so what does that mean for jygallag and sheogorath.
We'll have to wait for a sequal to find out.
5. Will it be written in lore that the champion of cyrodil was also sheogorath.
I'd imagine so, since that's what happened.
6. Is haskill immortal like jygallags trapped chamberlain guy.
Haskill is a Daedroth, and all Daedra are immortal.
7. How did the other daedric princes transform jygallag into shegorath.
They're all gods. And there were over a dozen of them and only one of him. You do the math.
8. Your personnel opinion on the future of sheogorath/jygallag/ the shivering isles.
I'm not convinced that all that stuff about Jygallag being cursed into Sheogorath is actually true. After all, we only have Jygallag's word for it, and he's a form of Sheogorath. Sheogorath is the lord of all mental illness, and Obsessive Compulsive Disorder (which is what Jygallag basically seems to be the embodument of) is a mental illness. So is split personality disorder (one person turning into another person then back again) and paranoid schizophrenia (thinking your neighbors, in this case the other Daedra Lords, are conspiring against you.)
So my personal interperetation is that everything we saw in "Shivering Isles" was part of Sheogorath's delusions. I ain't a dev', so don't take me seriously, though.
If I'm wrong, then I bet that Jygallag will eventually return to the Shivering Isles and try to "order" it again. Since he's a logical fellow, he'll probably be back with a better strategy, and it'll probably be [censored] awesome. Can a brotha' get a multi-realm war between every Daedric Lord?