Let's say I've got a 5970 and a 5770. Obviously these two cannot be Crossfired, but for one thing, they can still both mine on their own, even if they're on the same motherboard, correct?
Correct. Even more, there are numerous reports that mining separately is better than CF (more Mhash/s).
Also, as far as gaming goes, what if one of these cards was run as a 'dedicated miner' while the other was plugged into the monitor, either not mining at all, or at low aggression. Would gaming at a decent framerate be viable with this config?
Yes. Of course, it does depend how much RAM and CPU the miner needs, but that is not usually a problem.