I always permanently remove the back plate on my floyd-rose equiped guitars. It just saves time(and who's gonna pay attention to the back of a guitar anyway? except for us gear-freaks, lol), and I'm pretty sure that's the reason Steve Vai does that too.

As for the springs coming loose, I've never encountered that problem myself. But, I would try wrapping an elastic rubber band around where the springs are attached, that might just work.