The punch style skirts aren't a necessity. They came from the popularity of throwing jigs into really heavy cover. The problem with a jig is that you tend to hang if you're working in really thick stuff light mats. The skirt gives you the bulk of a jig, with a fairly weedless bait if you hide the hook in the plastic.
The barbs on hooks are primarily there to hold the plastic on the straight shank hooks. You can also use a toothpick run through the plastic and the hook eye to do the same thing. Make them yourself if you want to use the barbs, it's far cheaper. Gamakatsu released a hook at ICAST that I've been playing around with making my own for a while. It's basically a metal barb heat shrunk to the hook shank instead of just the heat shrink pulled into a barb shape. It definitely holds a little stronger.
Weights:
Use something big enough to get you through the surface layer of the mat. After that, you may need to shake the rod tip a bit to get the lure completely through the vegetation. I like tungsten primarily because its much more compact, and it is far more sensitive than lead.
Hooks:
I like using a straight shank hook for flipping. Really, I like it for all plastics. The hook up ratio is much better with them. Even more so if you snell the hook. Lots of guys like the Mustad Flipping hooks. I use the Owner XXX straight shanks. Take your pick. As long as you buy a good quality hook you should be fine. Size all depends on the lure being used. One hook that I've really had a lot of success with lately has been a Gamakatsu Octopus hook. You can get a hook with a large bite in a small package. It's great for smaller baits. When you snell it properly, the hook will shoot out to the side on the hookset.
Lures:
Really any soft plastic will do. I prefer lures like the Gambler BB Cricket, Sweet Beavers, Rage Craws, various creature baits, and other compact plastic lures.