After Shivering Isle questline, is CoC now mad and crazy like the old Sheogorath. Does he have to be crazy to take on the new role?

Its implied through a quest in skyrim that he became the exact same as the old sheo just with the memories of the coc. Part of mantling a god i suppose

Well, since Beth decided to be very obscure in TES V: Skyrim, there is no knowing whether the CoC became Sheogorath permanently or not. It has been confirmed by in-game events that Jyggalag has been freed and walks through the planes of Oblivion once again. There are many theories about the CoC and Sheogorath mystery.

1.(the theory i support) The CoC indeed become Sheogorath and the curse of madness slowly turned him into what he was shown to be in Skyrim.
2. The Sheogorath in Skyrim is not the CoC. Jyggalag became Sheogorath again after the CoC passed away or vanished after some time.

And some other theories...

What are the evidences to support this theory? How can CoC die since he's a daedric being now?

CoC "grew to his station" much like Haskill aluded that he would at the end of the questline.

Sheo in Skyrim is indeed the CoC since he mentions so much stuff that the hero witnessed fist hand.

He seems to still retain his memories from his old life, given how he's still fond of his friend Martin, talks about the brotherhood,the butterflies when he first entered the isles, Mathieu's head, the Gray Fox,the blood of the aedra/daedra, the uh, cheese... He also says he saw it first hand when Martin turned into a dragon.

"But my Oblivion PC was a female lizard, not an Imperial old man!" - It doesn't matter, Daedric Princes can take any form they may see fit. As CoC grows more into his/her role of Sheo, he/she probably felt like taking old Sheo's signature appearance was the natural thing to do.

It's not unheard of for lesser creatures to become Daedric Princes, it happened with Mehrunes Dagon and Molag Bal before.

Sheogorath does not call Martin his friend or say that he is fond of Martin. He says, to Pelagius, "You are the best Septim that's ever ruled. Well, except for that Martin fellow, but he turned into a dragon god, and that's hardly sporting..."

Given that Sheogorath is a god, it is entirely possible that he could be aware, as everyone else probably is, of events that happened 200 years ago.

The clues are there to be interpreted, since there was no reason for Sheo to mention the events of Oblivion in such detail other than Beth expecting us to speculate that he is the CoC. The quest in Skyrim could have carried on just fine without the references to CoC's past.

Anyway, he also says he "was there for that whole sordid affair",talking about the Oblivion Crisis.

Hm...hard to say. It could be you eventually became mad after taking on the role of Sheo, but I noticed some hints that imply that you become mad as soon as you pass through the gates of madness. (The Blessing of Mania/Dementia you receive)

So was the entire population of the IC, and to some extent all of Cyrodiil. Notice he never mentions anything only the CoC would know.

Does the entire populace of Cyrodiil also know about Mathieu Belamont's severed head, the room with the butterflies, the Gray Fox, the gathering of the blood of Aedra and Daedra, the cheese rain?
Anyway, he says he was there as if implying he was really there in person to see everything, and that he had a marvelous time while at it.

Anyway, my point stands, there was no reason to mention specific events from important quests in Oblivion if we weren't meant to make the connection between the CoC and him.

He mentioned a severed head. There are plenty of other severed heads in the game. My favorite ones are the ones stuck on pikes.

Yes. It seems like everyone knew about the Gray Fox. Blasted thief.

He only mentioned blood. Unlike "Fox" it wasn't capitalized, so there's no way to confirm this was what he was talking about.

Ditto. I read someone speculating that he was actually talking about the Ancestor Moths.

More of Sheogorath's monkeycheese lolrandom humor.

Of course, I'm not denying that we were meant to make a connection with some of these. There's enough evidence to make his ramblings mean something if we want them to.