OK...I"ll have to go down to the garage to get all the specifics, but Bianchigirl, yo'ure correct, it mounts on the dropout.
The person who described it as swaged might be correct too -- there was a small soft metal collar holding the stud firmly in place. I removed the collar (maybe a mistake) and now the stud spins freely in the hole it occupys in the hanger, but it still won't come out. At this point I think I may have backed myself into needing a new hanger that will solve my problem - since I suspect if I get the stud out, that the hole it mounts in is not threaded and won't accept a new derailler.
The orginal shifters on this were some low-end SRAM twist grip shifters, and they weren't working well. My daughter couldn't shift the things. I took it apart, greased it, tried to make it easier for her to work -- and it basically was just worn out I think. I couldn't really find a replacement, but found a cheap deal on a SRAM derailler that was a bit better (SRAM SX4) and a shifter to match it.
I hate bikes sometimes.