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