What is the mean population within populated Zip Codes?  [This
aknowledges that many Zip Codes have no human residents].

I figured this out some years back, so this may not be valid today. 
But I recall that the average was a little more than 30,000 people,
although there's a lot of variance, as you say.  (By the way, that's
not too different from the average number of people in a telephone
areacode+exchange!)  There used to be roughly 33,000 valid ZIP codes
in the US, I believe.    This is all from memory, so don't take it too
seriously!

33,000 zips X 30,000 per zip = 990 million people
Well, on the face of it there is a problem, but then there are many
zips with zero population which should be culled.

There are about 43,000 zips and 295 million people today.  That would
indicate 6,860 persons per zip. However, because of the zero
population zips (big companies, U.S. government offices, etc.) this is
incorrect.

I'm guessing that there are maybe 25,000 populated zips, which would
indicate 11,800 people per zip.  I could be way off, though.