It's like a screen, really. A paper-thin screen, but one that you cannot see behind. Normally, it's like there's a light on the Oblivion side: you can see the shadows of the Daedric Princes, you can hear them, they can occasionally reach out and interact with you, but functionally, they're hidden behind that screen. When they appear on Mundus as themselves, you're not really seeing THEM, you're seeing their shadow.
The Mythic Dawn started taking a knife to that screen and cutting away chunks, thus allowing the Daedra to step through completely at will. Mehrunes Dagon came toppling through the screen, knocking it down- functionally, the screen was gone, and Oblivion and Mundus were pretty much the same thing. Then Martin goes to get the screen repair man, who throws Mehrunes Dagon back beyond the screen, puts up a new, far more strong one, and nails it in place far more securely.
This is one of the better ways I've seen this explained. Good anology.
To sum up the OP's original questions as to Daedric presence in Skyrim:
First of all, with a few exceptions, Daedric presence is at an all time low in Skyrim. Lower than Morrowind, even.
Now, as to the cases like Sheogorath and Dagon's presence in Skyrim:
Dagon's Dremora are confined to his shrine. The shrine serves as an invitation, of sorts. You can see this in Morrowind as well, where there are daedra wandering around Daedric ruins and Dremora guarding the shrines to the princes themselves.
The rest of the Daedra don't really manifest themselves directly on Mundus. And any time they do have power, it's always in the presence of their shrine.
I always assumed these shrines to the daedra are built with a degree of magick involved, and they are linked inexorably to the daedra they are dedicated to.
Meridia is another example where this fits. She only has influence once you bring her an artifact which is already part of her essence (remember that daedric artifacts have bits of the daedra in question linked to them), and even then, she doesn't physically manifest, she just talks to you and does a light show.
Her actions after that are by invitation, yours.
Sheogorath is the one weird case.
In Oblvion, the door to the Shivering Isles is for mortals only, and the reason it works is covered in dialogue with Haskill. Basically, since only mortals can pass through it, it serves as an open invitation to mortals to visit the Shivering Isles, which is not against the rules. Remember that Sotha Sil also visited Oblivion at least once, and Vivec did as well. Granted, they weren't exactly mortals, but still.
Sheogorath's bit in Skyrim is similar, an open invitation, AND an artifact that's linked to the daedra himself (pelagius's hip bone).