It's a lot more technique than strength. I've destroyed guys much bigger than me by having better technique, but I've also gotten my ass handed to me by a couple of much smaller, but older and way more experienced guys. The best thing you can really do is just talk to other props and learn the tricks of the trade (the "dark arts" as they're called).
I generally go for the power variants. For one, I don't have access to a real oly platform so I like to stick with lighter weights. But also, I'm not really using them for strength gains so much as speed and power. I go heavy, but I really focus on getting the weight from the floor to the rack or overhead as fast as possible.
As far as floor vs. hang goes, the explosive lift from the floor translates really well to rugby. Think about body position when rucking and securing ball, or even tackling. Get low, back straight, head up, and explode up into your opponent. Only real difference is that in the gym you're on your heels, on the pitch you're on your toes.
The hang clean and press, on the other hand, is very similar to lineout lifting. Actually, it's probably more like a power snatch from the hang, depends how fat your second rowers are.
There's two very different schools of thought on this question, depending on what level rugby you're playing. First, for higher levels, sure, slow aerobic conditioning will help build a good base to work from. Won't necessarily translate well to the pitch, but it will help with recovery between bouts of heavy effort and sprints. I usually plan on doing some form of long, slow cardio during the offseason myself, although this year I'm nursing a bad ankle still.
Second school of thought, for the lower level club grades, you're a prop now, not a back. You have to learn how to think differently (more slowly?). Voluntary running? Only if there's another pub down the street who calls last call a few minutes later than the one you were just kicked out of. Just broke through the line at your own 22 with just the fullback to beat and feel like you can probably score? Run sideways until one of the backs catches up (it won't take long) and let them sprint the length of the pitch, it's unbecoming of fat guys. If by some odd twist of fate you find yourself with the ball sprinting for the try line, about to touch it down in glorious fashion... double check for the posts, it's probably the 22 and you're probably about to look foolish.