6 speed to 8 speed

Your old mech will probably work with friction shifters on 8sp, providing that the number of teeth on the sprockets are within the derailleur's capacity.

Yup.

I'm not sure about using 6-speed shifters on an 8sp cassette - the sprockets are closer together than what the shifters were designed for.

Yup.

You could just buy a 6 speed cassette and use it on the 8 speed freehub, though.

That won't work. 8 speed is wider (or at least 8 speed shimano cassettes are wider than 6 speed), you can use a spacer to fit a 6 speed cassette to an 8 speed freehub but the spacing will still be set for 6 speed and so, 8 speed shifters won't index properly.

You also have to make sure that the shifter has the correct cable pull ratio for the derailleur to work properly.

If it's all campag it should be fine.