Tvs have isolated powersupply, they have done ever since they added things like AV inputs and headphone sockets since they are exposed. Prior to that they were like older arcade monitors with the chassis referenced to the AC input, and just isolated the antenna input thru a capacitor or a small transformer. the transformer WONT make is safer to handle since you will still have a ground connected to the tv via the signal cables, so there will still be voltage in their referenced to ground.
No idea about the drill, they have some pretty mean magnets in them. Perhaps use an extension bit if you can?
The cord to the annode should be ok, because it is insulated, but I still don't trust it with all the caked on dust that they get.
If you are worried about handling it, go to a service technician and ask them to discharge it. They would like some liquid gold in return for this, pick up another tray of it for yourself when your completed the cab as motivation.