Last I heard, the next Xbox (I refuse to call it the 720, such a horrible name...just call it the NextBox or something, media!) is slated for a release in the Holiday 2013 window, with companies like Treyarch, EA and LionHead (hey, now that Peter Molyneux's gone, maybe the next Fable WILL be as good as promised!) working on launch titles. Pure speculation, obviously, but that's what I've heard about it.
Given that it's Microsoft we're talking about here, I don't see them keeping the 360 alive for very long once the next Xbox launches. There would be no incentive for the casual gamer to purchase the next Xbox if they kept the 360 around, so as a company, the logical move is to push the customer towards the newest product. I could see the 360 lingering for a year at the most, but not beyond that, unless the next Xbox is such a colossal failure.
As far as completely losing the physical me drive, I have to agree that it's unlikely for this generation; however, I wouldn't put it past Microsoft or Sony to make that move. They're doing all they can to push consumers towards digital goods as much as possible, and the next logical step is for games to lose the physical counterpart. If there is a physical media drive on the next Xbox, my guess is that within 2-4 years Microsoft has most of the major developers releasing games via Xbox Live Games on Demand the same day as their physical copies, so that fewer customers buy physical copies and they'll tail off to the point that it becomes a "logical step" for the companies to take. Cheaper for Microsoft and Sony, cheaper for the developers, etc. It'll anger gamers everywhere, sure, but the casual gamer won't care, and that's why it'll probably happen at some point. I dislike it, because I like to know that I actually own the game I paid for, but I suppose I'm equally hypocritical because I enjoy downloading games from XBLA (indie games, particularly). This is turning into a ramble, so I think I should end this here, but essentially -- I don't see the physical drive slipping away at the start, but wouldn't be at all surprised if it did in the later life of the next console(s).
Having said all of that, I did recently read an article that mentioned plans for an Xbox set top version, which is essentially what we're talking about there -- a stripped down Xbox good primarily for TV and apps, but with limited XBLA capability. Won't be the standard model, but a cheaper alternative for those who aren't big into gaming.