Yes, that is the hanger. On most better quality bikes, the hanger is actually part of the frame. In your case, the frame doesn't have a hanger, so your derailleur has what's called a "claw"
By the way, your wheels should be loosened then pulled further back into the frame dropout. Looks like it's about to fall out.
Originally Posted by
rule30
I'm going to go with #3 as it looks like the low limit screw is actually missing. If screws are falling out and things are a bit rusted then go ahead and take it by the shop. Its a lot cheaper than dental work and you'll need another screw at the very least. If you have already remove the derailleur make sure they take a close look at the spokes as there is bit of bend going on there too.
Good eye.