Ok, I got everything connected up. It wasn't without its hiccups, but I worked through them. Somehow, magically, the wiring between my Z stop switch and the DuetWifi board went bad. I tested this with a multimeter and proved that it was the wires running from the printer in the main bundle of wiring. I spliced in a new length of wires between the Z stop switch and the board and now it works fine.
I'm left with a single issue: I'm getting an error message from the bed heater. I've tried both orientations of the pins for the bed temp sensor, and the behavior is the same either way (I don't know enough about the temp sensor to know if there's a polarity or not.
I'll post that in another thread so that it gets new eyes.