I got a new Melzi and Nema17 from 3dnielsen.dk. The motor didn't budge when hooked up to the old Melzi, so it does seem to be the Melzi (at least).
I carefully marked the existing wires using masking tape and unmounted the old Melzi. Admittedly, it was a rather crappy cheap make, not silk screen or anything.
Wired up E Temp, Power, and E Motor, and tested. Didn't work, of course, without Hot End wired. Added then, then tried again. It responds to input (it has an actual Blinkenlicht), but doesn't heat. Oh, right, haven't uploaded the right firmware. Do that, adjust baud rate, trying again, and now it heats! Ah, toasty!
That's not enough to satisfy the extruder motor, though, it still refuses to budge. Try wiring it straight into the board, and that works! So there is definitely a problem in the wiring. Wonderful. Enough for now, though.
It seems to heat faster, so I have that going for me, which is nice.