Direct mount vs. hanger derailleur
The derailleur in the first photo doesn't appear to be mounted correctly. See how there's a big gap between the axle and dropout slot? I think you should just go with a regular, bolt-on derailleur. Non-replaceable hangers are usually found on steel frames (check yours with a magnet) and are perfectly fine to use. With steel, you can just bend it back into shape if broken, or even cut it off and weld on a new one if i remember correctly. Any 7 speed (it might even be 6/7/8 compatible) derailleur should be the right one.
The chain skipping could be caused by worn out drivetrain components like you mentioned, or it could be due to the wonky derailleur setup. I would wait until the correct parts are installed and it shifts fine before sorting that problem out.