Alright, so I recently read the book "Myths of Sheogorath", which is amongst other things about a king. Now, I found two particular sections in said book that made me go "wut":

Sheogorath heard their pleas and decided to visit King Lyandir. He appeared to the king in his dreams as a field of flowers, each with arms instead of petals and the face of the Madgod in the center. "I am Lord of the Creative and Lord of the Deranged. Since you have no use for my gifts of creativity, I have decided to bless you with an abundance of my other gift."

And then, later on in the book, (Granted it's a different section of it) it says this:

After three days, Ravate was convinced that every plant and animal was a tool of Sheogorath. He hadn't eaten or drunk for fear that Sheogorath had poisoned the food or drink. He hadn't slept for fear of Sheogorath invading his dreams. (Which was foolish, as dreams are the domain of Vaermina, may She grant us Restful Sleep.)

I do realize that these are myths, but that doesn't necessarily mean it's fiction. So the question is, can Sheogorath actually invade dreams, as suggested in the first excerpt, or can he not, as the second excerpt suggests? Just curious.

Madness-inspired dreams are his, I would say. That's a bit circular - are the dreams mad because he's in them, or is he in them because they're mad? He might have been able to invade other dreams in this due to some agreement with Vaermina. Of course, being mad, Sheo loves a bit of contradiction so it makes sense that he would include both an idea and its negation in his myths.

I believe that all the Daedras can pop up in dreams of dreamers if the dreamer have some affiliation with them.
For example the King being mad, or similar.

But that is just my thoughts.