In researching this answer, I found a job that I would like
to do almost as much as I like writing trivia. Any mail
that is improperly addressed or otherwise undeliverable is
sent to a "dead letter" office if the outer envelope
doesn't have a return address on it. There, a Postal
employee opens the envelope and checks out the contents.
I'm so curious, I would love to open other people's mail
for a living.

If there's no indication on the inside of the letter of who
sent it to whom , it is immediately destroyed. If the
envelope contains anything of value, the contents are
usually retained for about 90 days in case someone is
trying to find them. Any unclaimed items are auctioned off
to the public, and cash is placed into a general fund.
Money that remains abandoned after 1 year is rolled over
into a general account to be used by the Postal Service
however they see fit.