Direct mount vs. hanger derailleur Currently repairing a bike that I got off the road for free years ago. Around three years ago, a stuck link caused a chain jam while pedaling at high speed, causing the metal around the b-knuckle on the rear derailleur to shear, rendering it irreparable.
Currently have gotten into biking again and am trying to repair this bike by first cannibalizing a derailleur from another broken bike I have. I got the shifting to work relatively well, except the current derailleur is bolt-on and the parallelogram gets caught on the derailleur hanger when up or downshifting between gears 4 and 5 (see picture).

The derailleur hanger, for some reason, is not removable unlike the others I have seen online while researching. Thus, I want to buy a derailleur that mounts onto the dropout. I've already found the tooth capacity I need [(48-28)+(28-12)=36], but now I am not sure on the mounting method to use. If the dropout looks like this, should I buy a hanger or direct-mount derailleur?

For reference, I would have bought one to match the still functioning front derailleur, but the model, a Shimano Exage 300 LX, seems to be out of production (and used ones are a bit too expensive, I'm looking for something &lt;$40). Any recommendations on compatible derailleurs to buy?
Also another issue I've noticed is sometimes the chain &quot;skips&quot; on a single gear under load (not between gears) What could be causing this? I checked every link individually and none are stuck. I've searched online and only found a solution where the guy said he replaced all the chainrings and cassette without mentioning what the problem might be. I don't want to replace anything outright if I don't have to, any ideas?
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.