Good news! I started digging into the camera, and found what was causing the trouble.
Ensign Light Leak by
Berang Berang, on Flickr
The light leak was caused by this old piece of tape which has come unstuck. It shields the mirror pivot from light entering through the ground glass.
Ensign Light Leak by
Berang Berang, on Flickr
Not a good thing!
The shutter is actually far more complicated than I assumed. There are lots, and lots, and lots, of moving parts. Also lots of sliding surfaces. I ended up using a q-tip to thinly coat a lot of metal to metal surfaces, this resulted in the shutter being smoother, quieter, faster, and oddly enough removed a lot of the mirror slap.
Ensign Mirror Box by
Berang Berang, on Flickr
The problem with the shutter not exposing turned out to have to do with this little latch and tab:
Ensign Shutter by
Berang Berang, on Flickr
Ensign Shutter by
Berang Berang, on Flickr
The tab with the cut out in it (to the right) is connected to a shutter blade that caps the shutter after the mirror swings up. The latch is supposed to hold it down, and thus hold the shutter open, until released by the pin on the left. The mirror hits the pin, which releases the latch, which allows the tab to spring up, which closes the shutter. The problem was this latch wasn't latching reliably, so sometime the shutter would close before being triggered by the mirror. The solution was to just bend the tab slightly down so it would catch on the latch sooner when resetting the shutter. Works fine now!
Ensign Shutter by
Berang Berang, on Flickr
Ensign Shutter by
Berang Berang, on Flickr
And here you can sort of see how it works:
Ensign Shutter by
Berang Berang, on Flickr