Sounds like fun! What item are they on? Might lead to some more clues. If it's something relatively common I would suspect it's number 2, number 1 sounds fairly complicated for any kind of mass production.
You might find a set of cheap "screw extractors" might help, there's a few kinds if I remember correctly, one that requires a pilot hole drilled and some that don't.
Failing that, drilling it out is my usual approach to getting at electrical goodies, especially for rivets (in fact I think I did this recently for my ham radio mast thing at the space).Your goal really is to separate the head off the screw, and that usually works quite quickly with the judicious application of the correct drill bit size!