I got my DS back today, good as new! It looks like they just replaced the plastic case and wiped my settings, since Tetris didn't complain about a different serial number when I played on the internet. All told, the repair process took just a bit more than 2 weeks. FedEx took 5 days to ship to Nintendo. Nintendo took 11 days to repair and ship back to me.
The deal is that you get one hinge replacement for free. The next time it happens, you have to pay to fix it.
It was inevitable, but my Passcard-3 has died, so I won't be able to reflash my DS again. I suppose it's about time I got an R4 anyway.