5-speed freewheels do exist, but are getting harder to find since the push is for more and more gears. As long as you replace the chain when it's worn, the existing freewheel should last a while as they're quite robust. (There are no 5-speed cassettes- cassettes came about when the length of unsupported axle became too much for an 8-speed freewheel)

A 7-speed conversion may be do-able, but will require widening the rear triangle to suit the slightly longer axle. A steel frame can be slightly bent in this way without problems. Using the original wheel with the new freewheel will require the wheel to be redished and spacers added on the right side. Alternatively you could try to source a pre-built wheel with a 7-speed freewheel or cassette.

If the bike has friction shifting, the limiting factor is the derailleur. If it can cover the range of a 7-speed freewheel, then the system will work with the old shifter. A modern freewheel should be easier to shift due to the shaping of the teeth.