My approach, with a DSLR setup with dual DS-125s:
The camera rig is weighted as close to neutral as possible, erring to slightly negative, and it's even trimmed so that the optical axis is close to horizontal.
There are no clips on the camera, instead there are four about 1" diameter loops in each corner made of cave line pulled inside silicone tubing - the tubing keeps the loops nicely open all the time, and easy to clip to. If I need to clip off the camera, it's done using an extra double ender or a dedicated "leash" made of a bolt snap tied to a D-ring with about a foot of line (again pulled through tubing to prevent tangles).
I usually don't clip off the camera. As it's neutral it's no big deal to carry it through the dive, I've done hours at time, no biggie. If I need to do anything involving 2 hands (signaling in dark comes to mind) the camera stays put if I just let go, long enough for a stage switch in cave, for example. In a current it simply gets temporarily clipped off to whichever D-ring is free, or handed to a buddy. If scootering, the camera sits on the scooter. About the only time I clip off the camera for longer periods is during deco ascents on an upline - then it gets clipped from the scooter ring using the leash, so it hangs a bit below me and doesn't get banged by tanks.
Then again, if I take a camera with me, it's a photo dive - the rig is simply too big to take just in case. So things get planned for it being along. I can see how clipping off the camera could be tempting if it's just along if anything photogenic shows up.
A few times it's been along just to take wreck identification photos in case we find something that might give an ID, but I've been looking into a smaller P&S for that to tuck in the dry suit pocket. Haven't found a small enough package with a decent depth rating yet. The Gopro range might fit the bill...
I've been thinking about an extra pocket for a dome cover, but haven't had a real need for one yet. Would be nice to have the dome covered when handing the camera to a boat, though.