Discovered one vent (so far) where the "box" inside the wall was
simple cardboard. The furnace is probably 12 years old, ductwork is
probably older than that. The duct itselfs comes up about 2" short of
actually contacting the cardboard, meaning there are gaps where I can
see directly into the basement.

Was this some sort of normal hack years ago, or some nimrod's idea of
a cheap shortcut? It certainly seems dangerous to use paper products
around a forced-air gas furnace.

Is it feasible for an average handyperson to construct a new "box"
with duct board or sheet metal & mastic? I have tin snips & other
basic tools, although no way to make decent bends if the sheet metal
is too thick. Duct board is new to me. I've worked with drywall and
greenboard, however.

Thanks.