I remember reading a post a long time back, in which it was stated that Dragons were not something to be slain in the TES universe, that they were more a creature to be revered - almost as demigods. This sat well with me after seeing the end of Oblivion. I'm extremely interested in hearing what brought their sudden reappearance about, and why it's now fine to slay a dragon. Maybe it's something to do with the Septim line ending?
I think it has to do with the fact that they're now trying to destroy everything, myself, which in turn is because Alduin is trying to eat the world. I think the real question is why Akatosh, who is supposed to be the same being as Alduin, previously seemed to be protecting the Empire but now has more destructive intentions, hopefully, that will be explained in Skyrim.
'm thinking they'll evaporate after defeat. Especially seeing as you absorb their soul or whatnot.
Why would that make thme dissappear? Creatures in past games didn't dissappear when they were soul trapped, so why should absorbing the souls of dragons cause their bodies to vanish?
Anyway, my guess is that the skeletons of dragons will behave the same way as the corpses of other creatures in the game, which I'm assuming is the same as in past games, linger until the cell respawns, which in previous games was generally three days. Though I guess I could be wrong.
As to dragons after the main quest, we don't know, but I suspect that once the main quest is completed, the player will encounter no more dragons, obviously, though, that's just a guess.
In OXM it said there is an animation where the dragons burn up or something... I'll go dig out my copy and find the quote now for you.
You can already see something like that at the end of the gameplay trailer, I'm guessing they were describing that.