I agree, the cogs don't look worn enough to be causing problems. Anyway, when teeth are worn the problem usually presents in the smallest cogs first, rarely on the largest. Judging by the photos I think your derailleur isn't adjusted right; back out the lower limiting screw and the chain should settle down on that big cog.
Should. Yes, I know, I said it
should.
What shifter mount is that? I am not familiar with Campagnolo Nuovo Record stem shifters. Am I right to assume these are the standard downtube shifters attached to an aftermarket mount? If so, then that --specifically, the interface between mount and shifters-- is probably the problem.
I'd agree. I'm not sure that Campy ever made a Record stem mount shifter, but the bosses on that mount look wrong. The lever does not seem to be seating far enough on them. For reference, see this pic from velobase
