Well, I returned the laptop to the university.  It turns out the person who
had it before me had the same problem of OS X not booting sometimes (with the
original disk, not the new SSD we installed when I got it).  When I got the
laptop, I was in fact told that it "wouldn't start", but we understood that to
mean "wouldn't turn on".  So when it *did* boot from the OS X live system we
supposed that it wasn't in fact completely broken.

Judging from the nature of the problems I would guess the firmware (EFI) is
"broken".  Perhaps reflashing it would solve the problems, but the laptop is
old enough that the university will probably just buy a new one instead, what
with the battery being broken and all.

So thanks for the help, but the laptop in question is pretty much officially
finished.

Greetings
-- 
Marc Joliet
--
"People who think they know everything really annoy those of us who know we
don't" - Bjarne Stroustrup