On a mandolin should the meaty part of you hand below your palm, gently rest on the bridge or the strings behind the bridge, or should it not touch at all?
Most of the players I've watched play with a floating pick hand. I pulled up a Chris Thile video and am seeing that maybe he rests his forearm on the edge of the body but otherwise floats his picking hand. If you want to mute the strings like a guitarist would, that's fine, but for most cases, it's probably better if it doesn't touch at all.
I dont' think resting your hand behind the bridge is too bad - a lot of the overtones produced there aren't desirable anyway, and lots of mandolinists use mutes on that side to mute these overtones.
However, if you have pretty small hands/arms (as I do) you will find it is hard to pick closer to the fretboard without muting in front of the bridge.
It is almost always best to have as much flexibility and potential for motion as possible, and planting any part of your picking mechanism is going to limit this.
I use initial contact on the bride on the base of my palm to establish reference, but then I float as much as possible, as VarLogRant mentions.