Welcome back to the Shadows and welcome to Anarchy!
1) Yes, those are Logic and/or Agility-related Tests.
2) Stun damage does not, strictly speaking, translate into Physical damage. You can try it and see if you like it (I have not found it necessary). Once your last Stun CM box (or circle) is filled, you are Staggered and are pretty much down. If you get hit with more Stun you are Knocked Out and are definitely down but not dead. I posted some GM Screens/Narration Aid earlier, it has some alternate rules for KO/Death that you might like.
3) Yes, Amp for the drone and burn a weapon OR take a weapon Amp like "Drone/Vehicle-mounted Machine Gun". Whichever you want.