I think it's an interesting idea and I would be on board for having a go at a remix.
Personally I would prefer a song that already had a finished mix and could give me something I could compare my own effort to, as well as hear what others have done to finished tracks.
I would have no problems with altering the format of the song, and any or all alterations would be acceptable if this was agreed upon. I think the judging could take into account whether alterations to lyrics, rhythm, melody, etc. would contribute to a better mix.
If recording the tracks was part of the "competition", then I think this would add an extra layer of complexity and could be a separate competition if required.
Craig has already posted that he would be happy to provide the song for a remix, as I'm sure others would if needed.