Mac help received!
Well, flamingatheist came over this afternoon with a USB DVD drive. Before attempting that, he took off the back cover, fiddled with a few connections, twiddled this and diddled that...
And the thing started again once we replaced the cover.
I repaired permissions (suddenly this thing is speedy again - it took about 20 secs.) and so far so good.
What did he do to it? Neither of us knows. He chalked it up to the "PFL factor" - a technical term where the P stands for Pure and the L stands for Luck.
Thanks to everyone for their suggestions! The DVD drive is still shitouttaluck, but at least the computer is rollin' again.