Andy,. last year i had this same problem with my Ebay rescue Panzer III. I don't have any advice on where and what to solder as i had the same issues, and more, tried to fix the same things the same way and failed. My advice is check the numerous wiring diagrams on the net as to where the wires connect to, try and resolder, rewire etc. give this 3 or 4 more attempts to fix it. If it still doesn't all work then give up on that upper hull and either replace the entire thing or scrap it! I tried fixing my upper h ull electrics piecemeal and nothing would fix it as things just quit working one after another, or worked intermittently. Mine was a $35 ebay tank, worked for a while, then started going buggy. First the turret stopped functioning, then the gun elevation, then the lights, etc etc.!
I fixed all this in early 2014 and first I tried the hard route -much as you did, one circuit at a time. I used a volt meter, tracing the circuits, changing wires, and trying to resolder connections etc. came to the choice of toss it or fix it right so i chose the latter. Really the easiest and probably cheapest solution, I just bought another upper hull. Naturally ebay probably the cheapest source, if you buy another used panzer III at least you'll only have to redo the upper hull. If one doesn't come up on ebay in a while, here is the best place to get a brand new upper hull with everything connected and working, rcparty.com.
They're a legitimate company i checked them out before i bought from them, and have bought so far the mark III hull, a jagdpanther upper hull, some metal tracks for the Mark III ($50 shipped), they're on the level.
http://www.rcparty.com/en/panzerkampfwa ... 8-018.html
You can see the price, in US dollars, the shipping was $17 more, i looked at what i paid it was $40 shipped for the upper hull. Again, certainly it's cheaper and faster to just find an ebay Mark III for that amount if possible. But if you just want an all new upper hull, this piece does come with all the electronics, (note this piece is just the upper hull part of course, it includes the hull, the turret, the headlights, both the turret motors etc. all hooked up ready to plug in to your Rx.) already installed and mine worked fine right out of the box, no problem.
A couple important things, if you do order from them: the first shipping quote you get for it, is going to be jaw dropping expensive - but dont worry! There is an "epacket" shipping option you just have to keep clicking til you find it! Important item number 2, their shipping is pathetically slow. Mine took almost two months to arrive. But when it did arrive it was perfect - like new part and like i said, it all worked and still does. Please keep us updated on how this little tank turns out!