Thanks for the insight. I have replaced the cable with a different type and replaced the heater+thermistor. The new heater, connected to the E2 heater output, now heats. The hotend fan, connected to FAN2, also works.
Unfortunately, the bed heater seems dead, too. This makes me think that it might not be the wiring of the effector that caused this whole mess. But who knows.
Unfortunately #2, I can't get any thermistor channel to work either. The temperature hovers around 30-31 degrees, varying slightly, which makes it look reasonable, but it never changes when the actual temperature rises, and the firmware faults after a few seconds. It even stays the same temperature even if I completely disconnect the thermistor wires! (Perhaps the bed thermistor does still work, but I can't tell since the bed never heats up anyway.)
So I think this Duet is pretty useless, and I need to buy a new one. Thanks for all your help!