Just in case anyone else suffers the same maddening problems, I’ll post my (hopefully) final update.
After exhausting all of the troubleshooting, I started reporting every crash with Apple (via phone). They set me up with a Genius Bar appointment, during which they sent my Mac in for repairs. They replaced the motherboard.
Things seemed better initially, but then the crashes started up again.
Another series of phone calls, and another Genius appointment. This time they replaced the motherboard (again) and the card with the power and USB ports (sorry, don’t remember exactly what this was called; I can look it up if it’s important).
Since the second repair, it’s been stable.
So, in the end, it was hardware. Thank goodness I paid for Apple Care (though if I had pursued this aggressively when it first started happening, I would have been within the first year of coverage).
Thanks for everyone’s help.