In terms of the pressure relief valve...
... after the events described in the first post were followed by some dubious diagnoses and actions, which ended up solving the problem mostly by chance, I asked around locally for plumber recommendations, and received one. I emailed him the same description as in the first post, and his reply included this quote:
I just recently worked for a customer that had a relief line that leaked because an elbow was never soldered. Everything is hidden behind drywall and rarely has water in it so a missed solder joint is actually never tested. I think her home was 20+ years old by the time it actually leaked but it ended up ruining her wood floor in the living room.
So,relief valve is passing water, no water at end of discharge line, and water appearing beneath water heater in water heater closet...
PS The answers in this thread were a
lot closer to the mark than the diagnoses of the guys who originally showed up at my house (which is what I was hoping for, given all the good info I have received here in the past).