While playing Oblivion, a couple of questions just sprang to mind.
If the Dragonfires keeps the Daedra out of Mundus, how:
1: Can Sheogorath open the portal to the Shivering Isles? (Unless it takes place during the main quest, canon-wise)
Yes. The Portal is not a gate that allows Daedra in the Mundus, it's the exact reverse -- a gate that allows mortals out of the Mundus. Thus, it is not blocked.
As for the island itself, well, it's just an object, not a sentient creature; therefore it isn't bound by the barrier.
2: Can Hircine enter Mundus during Bloodmoon?
Yes, because it is a specific event that specifically allows him to enter. Also note that it is restricted in what it allows; Hircine can only manifest an aspect of himself, rather than being entirely present on Mundus.
3: Can Umaril threaten Tamriel? (Again, this is likely the same as the first one)
Yes. He is half-daedra and half-mortal, so he gets best of both worlds.
4: Can there be Daedra in dungeons and shrines?
Yes. They are summoned there by mortals; they do not come by themselves.
Also, shouldn't the player, upon becoming Sheogorath, be barred from returning to Cyrodiil? (I can see how this would be annoying gameplay wise, though.)
No. Like Umaril, the player gets best of both world thanks to his or her special, unique nature. In fact, the player is actually quite close to Umaril if you think about it; Umaril gained his Daedric nature thanks to Meridia's sponsoring, and the player thanks to Sheogorath's.