The 1st thing I would suggest is not to panic, if necessary take a break from trying to diagnose what's wrong if you're getting stressed out. Often while not thinking about the problem (having exhausted everything you can think off), an answer or possible solution will pop into your head. It's a bit like mislaying something, you check everywhere you can think of, including some highly unlikely places and you start to panic. If you take a break, maybe overnight, sometimes a possible location just pops into your head the next morning and lo and behold, you find what you were searching for.
I did dismantle the column switch on my own 600 (back around 1978!) - by drilling out the rivets and replacing them with shortened 'Pop' rivets when reassembling.
[To shorten a Pop rivet, drive the pin/mandrel out of the aluminium rivet, shorten the rivet then re-insert the pin/mandrel and squeeze with your rivet gun Nowadays it might be possible to obtain very short 'pop' rivets e.g. Lidl occasionally sell some suitably very short ones in a selection box for small money or maybe try a good Motor Factors.]
I didn't find anything wrong inside my column switch after dismantling it and would have achieved just as much by spraying some switch/contact cleaner into it and saved much time and effort.
So I don't think it's necessary yet to dismantle the column switch (nor buy a new one until you've determined it's faulty) - you can probably check continuity of the various wires and switch contacts by probing in sideways. Of course, knowing which contacts and therefore which wires are connected in the various switch position would be very helpful but I don't think you've found this information yet? Can you look in sideways to see the contacts that each wire is connected to and the moving contacts that bridge the appropriate contacts and maybe figure things out this way?
Someone recently uploaded? a copy of an original Fiat 600D Owner's Manual here - it's available free in the Downloads section. I checked this out and it seems to have a slightly better wiring diagram as regards the switch internal connections than the one you posted.
Also bear in mind that on old cars, especially one that may have been unused for a long time, that switches etc. can corrode, stick on/off, fail to make contact. Don't forget to check the dashboard switches. Also the ignition switch may have become faulty? Are all bulbs ok, do all lights have a good earth?
If you can't manage to find the problem, report back here and I'll study the above wiring diagram in the Downloaded Fiat 600D Owner's Handbook and try to guide you through a possible fault finding process for whichever lights are not working.
