I can't track down the source of chain skip - what might it be?
Does the large chainring show much wear? I don't think it is the culprit but it needs replacing when it is worn. If the old chain was very worn like that it could have caused wear here too. You could think about chain suck but IME this is quite violent and it jams up the drive by pulling back on the chain; if you look for it you can tell when its chain suck. Check it out, but to rule it out.

As you say the problem is bad moving off from a stop, I would strongly suspect the pawls of the freewheel mechanism in your hub as this is when torque is high on the cassette. I've had the same problem before, it's pretty terrifying when the pedals slip and you lose your balance in traffic. I had mine replaced and instant fix.

You did explore this with club mate's wheel, but I would be pursuing it again with another different wheel and you should at least get into the hub and clean, inspect, grease. Check the wear on the rim and see if it isn't time for a new wheel anyway, otherwise check out the cost of replacing the freehub. You've done the chain and cassette, good, but now you need to take it further. 

I think (from reading your post) you can judge the indexing and RD behaviour competently enough and this problem isn't super common despite all noisy, badly adjusted, worn out drive trains there are in the world.