My question is how can Sheogorath's realm still have a door to it in Tamriel when I have closed the Oblivion gates forever. Is there a difference between an Oblivion door and a gate?

Another thing, Haskill says that the door "poses no threat to Mundus." How can this be when Mehrunes Dagon's door's did pose a threat?

Only mortals can go through Sheogorath's invitation, daedra cannot pass through it. By contrast, the gates opened by the mythic dawn allowed Daedra through it. As such, sheogorath's door didn't violate the rules, whereas an oblivion gate like the one's seen during Oblivion's MQ would normally close within seconds to minutes of opening.

Pretty much what Darth said. Shivering Isles's door was more of a invitation. Mortals could go in and out, but no Daedra could come out of it.

There is a barrier between Mundus and Oblivion that stops exactly this:

1. Deadra from Oblivion without Invitation or knowledge of the First Secret.

Just about everything else passes effortless in any direction.

Not to be confused with Sotha Sils pact in which the Daedric Princes agreed not to be summoned by the lay.