I used a modern rear wheel and cassette with the original Suntour deraillers on an '84 Schwinn. There was enough movement in the derailleur and it shifted fine.

The wheels are 130mm wide though; yours is 120 or 126 if it's a six speed. You will need to spread the frame then bend the dropouts back so they bear squarely on the ends of the hub.

You cannot put a modern cassette on an old freewheel wheel. It's a totally different hub.

Not totally sure but I think your frame is compatible with any modern Shimano bottom bracket or crank unless it says "Nottingham" anywhere on it.

How do the gears stick? It could be an easy fix.