Yes, you need to remove the rubber rings or seals from the caliper body. You need to clean out the groove they fit into. This is one of the major points of doing all this. Over time, brake fluid naturally works it's way into those grooves and under the rubber seals. It dries out and starts forming a crust. Eventually, this builds up enough to push the rubber seal out more. This causes the brake to start sticking on, dragging, and binding.
As for the pistons, I really can't say. If that ring of corrosion is where they pass through the seals, as Fred said, then they may be no good. You'll have to determine if it is.