Putting spacers between the freewheel and the hub is not a good idea because there aren't all that many threads on the hub to begin with, and they are soft aluminum. Driven on hard by pedaling in low gear, the freewheel might strip them. Better to install spacers under the locknut on the *left* side of the axle (and remove an equal width of spacers from the right), which will move the hub and freewheel together further to the right, closing up that gap between small cog and dropout. Leave a millimetre or so between chain and dropout. This is usually achieved by having the outer face of the right locknut protrude 3.5 - 4 mm beyond the outer face of the small cog.

You will need to move the rim back to the left (by loosening drive-side spokes and tightening left-side ones) so that it remains centred between the dropouts. This makes for a less asymmetric, and therefore stronger, wheel because the spokes now have closer to equal tension on the two sides.

If this is too much trouble for the degree of sophistication you want to apply to this "old bike from parts on hand" (and it is a fair bit of work), best just to leave the wheel as is and live with the large gap at the right dropout. As long as the derailleur high-limit stop screw is adjusted properly (as mentioned by a previous poster) it will work fine.

(To optimize chainline, the smallest rear cog should be closer to the dropout than yours is going to be, but this is more important in theory than in practice. I mention it for completeness, since you were having issues with the front end as well, which also affects chainline.)