Is a heavy weight better to keep it on the bottom? It can be. For me I cast it most of the time, and I like to keep it in one spot and move the bait and not the weight. Some say that weight depends on water depth, but for me it depends more on wind. For an example, I fished a tournament one year that I had some smallies located on a hump. The wind was blowing 25-30 mph, not ideal conditions for a dropshot, but I stepped up to a 1/2oz weight despite only casting into 3 feet of water. Once the weight is on the bottom, the sink rate is only controlled by the weight of the hook and sink rate of the plastic. If you're hopping it, swimming it, or fishing it vertical the weight plays a more important roll as it controls how fast the bait descends.
What's a good distance from the weight? Depends on where the fish are located, but I usually stick to around a foot.
Can I fish ANY soft plastics or should I stick with worms? You can fish basically any plastic on it. I've had the most luck with 4" finesse worms, the dream shot, and the Strike King Baby Rodent
What size hook is appropriate? Choose a hook that fits the bait you're using. I usually nose hook the baits I use so I use a 1/0 VMC Spinshot circle hook the majority of the time.
Have you had better luck weedless rigging the hook or simply hooking it in the head? I usually hook the baits I use using a kind of modified nose hook technique. Instead of coming all the way through the bait like you would normally nose hooking, I stop halfway through the worm and bring the point out the front. I don't expose the point, I leave it just inside the worm. It keeps it more weedless/snagless, so I guess my answer to your question is both.
Is there anything against rigging two hooks? If you can get it to work and it falls within state regulations, I don't see why it wouldn't!
My best advice is just go play with it and tinker with it until you find something your confident in and works for you. You've got a great starting point from what people have already wrote in, but you can see there's a wide array of responses. Figure out which you like best and have a blast wackin'em!