Are you sure the derailer is mounted correctly? In-line with the chainrings, and rotationally around the seat tube? The fact that your low gear both rubs AND drops the chain tells me there's more than just cable tension and limit screw issues going on.

Personally, whenever I encounter a basket-case derailer, I start from the beginning and treat the problem as though I were installing the derailer from the beginning- unhook the cable, turn the barrel adjuster in about 75%, realign the derailer, check the low limit adjustment, connect the cable, then check the high limit adjustment and cable tension, in that order.