Re:
Possibly!
I assume you're talking about a 7 speed Synchro or Synchro II downtube shifter.
They were notoriously finicky even when set up correctly. The 7 speed shifters had a range of different coloured inserts depending on which rear derailleur you were using. There are charts available, but they don't mention which insert to use for more modern derailleurs like yours. Basically, trial and error.
The cog spacing on the Uniglide 7 speed is the same as other 7 speed systems, and the same as Campagnolo 8 speed. Campagnolo 8 speed shifters with the Veloce derailleur would work for 7 speed.
Another thing to note is that Uniglide had its own type of chain. I've used SRAM chains with Uniglide freewheels for friction shifting and they are fine, but I'm not sure how you'd get on with an indexed system.