The Corpse is an excellent idea, and I know it often works out surprisingly well. It's a more sophisticated and usually funnier variant of the 'everyone writes one sentence of the story' game, and I'd love to see it implemented. A word limit, however, would be vital, to prevent people overfond of their own prose (like me) from spouting entire novellas where others might submit only a few paragraphs, and therefore imbalancing the overall tone of the story.
I also agree with Astinus and bobandbill on the 'in-progress sharing' idea and the method for developing the CYOA story, respectively. It is possible to write a CYOA story by oneself, and I've done it before, but it wouldn't do much for community cohesion without some method of active participation. If the success of A Smell of Petroleum Pervades Throughout (which I'm thrilled about, by the way. Huzzah!) is anything to go by, there shouldn't be too much trouble with people not responding - and anyway, if people who had agreed to write the next segment pass their time limit, I'm sure there are others in the community who would be happy to take their place and dash off a quick chunk of prose to keep the story going. (For a start, I'm happy to write anything at all at pretty much any time as long as it can be dark, funny, or both.) We'd need to decide upon a method of selecting the next writer in the chain, though, and also a reasonable time limit for them to respond within that would be long enough to give them time to write but not so long as to hinder the development of the story.
As for prompt challenges... I think they're a good idea, but only if there's going to be some kind of feedback on them afterwards, perhaps from a rotating panel of volunteer reviewers. This wouldn't have to be an in-depth critique like the feedback usually derived from the SWCs, but if people don't get any response at all, I can't see that the challenges would last that long; it's just as discouraging as posting regular fics and not getting any replies. I think there'd have to be a fairly strict word limit, too, in order to keep entries short enough to be read and processed by others.
Actually, now I think about it, we could open up the prompt challenge entries for audience review, but I'm not sure how much of a response that would generate, given that the prevailing mindset in FF&W seems to be at best 'read fics and stay silent' and at worst 'post fics but don't read them'.