Derailleur and back wheel alignment issues
That's one seriously chopped-around bike.  I see a bolt holding the rear derailleur on, on an axis that would normally pivot.   This means your B tension is non-existent or infinite depending on the gear chose.

I also see the hanger well out of position and missing its other bolt.  That means the derailleur is at a funny angle.  Attempted `shopping below:

Additionally, what's that brownish dish thing that is about where the smallest cog would be?  Maybe its supposed to be there to stop the chain dropping off the littlest cog ??

I suspect your chain will also be too long, once this angle is all corrected.

The gear cable outer also looks kind-of lumpy, as though the metal reinforcement is kinked or broken.  This may cause issues, or with friction shifters it may simply provide vagueness while shifting.



The wheel is sliding forward in the dropouts because its either not tight enough, or there's not enough resistance.  I think the axle nuts look like generic hardware store nuts, and will probably lack any serrations underneath.  Some time with a file could restore rounded serrations back to grippier ones, but really some new axle nuts would be better.

  This is your image, simply rotated so its about normal angle.  From this you can see how far back the derailleur is.

Here are some lines drawn on top of your image to show the angle difference:
  That empty little hole in the hanger is missing a special bolt, which stops it rotating around the axle and undoing your axle nut and holds the derailleur on when the wheel is out.

This is a conventional claw hanger - there is nothing special here, these pre-date all the unique weird custom hangers.  These are worth around $5-10 of most currencies - it would be easier to buy a new claw hanger than mess around trying to make that weird nut on the back.



  This mess is a mochup of where your derailler should be sitting, relative to the axle.  This position means there will be more teeth engaged at once with the chain.



Another realisation just now - you have a stiff link in your chain.



This results in a noticeable click or clunky every 4~5 crank revolutions, as this inflexible link passes through the derailleur's jockey wheels.  This is probably the point that the chain was closed up at fitting time, though it can be caused by dirt.

To fix it, grab the chain in your fists on either side and try to bend the chain sideways.  Start with enough force to open a loose jar-lid, and work up to a bit more if the chain doesn't come free in that link.  



And finally - you say that the loose bearings are something to live with now.  I suspect this wobble is also contributing to the sloppy gears, because as the wheel moves it will move the whole cassette too.   I would totally recommend tuning up the bearings, and perhaps a clean and re-grease if you possibly can. 



In summary - I think this is fixable.  It will never be perfect, but I'm sure it can be better.