If your frame and hub are spaced 120 mm the most you can fit is an "Ultra 6" 6-speed freewheel. Sun Tour created the Ultra 6 to fit 6 cogs in the same OLD as a standard 5-speed freewheel and it required a "narrow" chain which is the same as current 7/8-speed chains. Unfortunately these Ultra 6 freewheels are now quite rare.

The largest 27" tire you can use will depend more on your frame and fork than the rim but 27x1-1/4" (630-32) tires were very common BITD.