Name of Blunt Tip Sheet Metal Screw

The fact it broke off means the screw is probably very stuck. I would first use a penetrating oil (though WD-40 might also work), and let it sit for awhile, before trying anything further.

Hindsight is 20/20 of course, but it would have been good to do this first. If you see any corrosion or rust, or suspect it's needed and have time, this never hurts to do.

You may be able to use a small drill bit to extract the broken extraction bit, and then try again -- but be careful as you may damage the screw beyond my next tip..

If you can't get the tip out (or don't want to risk trying), I would probably use a dremel cutting wheel, and basically turn the screw into a slotted screw. This can be difficult -- not deep enough, and you'll just strip the screw worse trying to get it out, and too deep and you'll weaken the head so much it'll probably snap off when you try to turn it.

The next thing beyond this is to use a "reverse drill bit" or "left-handed bit". You can drill right into the middle of the screw itself (with your drill in reverse), and the bit should actually come out before you've gotten all the way into it. Use a bit that is smaller than the thread size, otherwise you'll strip the threads so you won't be able to put another screw back in. Again, judgment is necessary here, too small and it won't get enough grip, too big and you'll strip the threads -- though maybe by this point you don't care that much :)

The paper is a key part of the drywall structure. Just as when you cut the paper on the drywall it's easy to snap, when you screw past the paper it's easy to blow out the back. Plaster and paper are a lot like concrete and rebar, the plaster based core of drywall resists compression, while the paper resists tension.

It's not essential to pull the old screws, the damage has already been done and will be mudded over. But you should add a second screw a few inches away anywhere you went too deep. This is a critical repair for drywall on the ceiling, and a very good idea for drywall mounted on the wall. I wouldn't go through any added effort to tear down the drywall, since it's still perfectly good. Just add the extra screws and be happy you caught the problem before experiencing a collapse.

Update: From your new photos, those screws are too deep. Once the paper has been torn, you lose strength at that screw location. You should never be able to see plaster around the screw head, but you also should never be able to run a flat edge over the drywall and feel the screw head above the drywall. It's a fine line to walk, but a professional drywaller should be able to walk it with ease (we train amateurs to do this in under a hour).

Also, from those photos, it doesn't appear that the installers are using a drywall bit (we refer to them as mushroom bits because of their shape). They leave a distinct ring around each screw, making it easy to countersink the proper depth, and preventing you from going deeper (the bit will cam out when it hits the paper).