1) I have a 12 volt DC supply. 12 volts seems to be
within the VIN range for the 7805s whose data
sheets I've now read. Can I simply apply 12 volts?
Yes, but that regulator might get mighty hot! I would
not do this for fear of cooking the poor thing.
That's what I figured to start with, before reading the
datasheets.
If as your photos (in another message) suggest it will run
OK from 8V with no heatsink on the 7805, then I would be
pretty sure it will be OK at 12V if you bolt a reasonable
heatsink to the regulator. Give it a smear of silicone
grease, of course.
-tony