How can I get a broken bolt out of the bathtub overflow drain?
Look at you lower photo.  See the pipe nut that is at the highest point in the photo?  That is a compression nut; there's a packing below this.

Loosen up that nut.  The tub overflow pipe (the thing you broke) telescopes inside that lower pipe.  The tub overflow pipe should now be able to go straight up until it comes out of the pipe, then you can wangle it down through the hole (or enlarge the hole).  

Myself, I just open a cabinet door, (giving access like in this blog here), and there you can see an example of what you're looking at.  Mine is in the Victorian style, a tasteful cabinet with a latch and hinges, and tall enough for easy access to the tub faucet. Great place to hide WiFi routers, personal stash, etc. Makes very short work of tub faucet and drain repairs. I recommend the upgrade!