DoomGaze

1. If you're playing on an emulator you can use the Doom Gaze Finder.
Otherwise, just cover as much area as you can. Doom Gaze doesn't change location as long as you don't land or go inside the airship.
2. Doom Gaze is weak to Fire and Holy, is vulnerable to Vanish, and has shell and safe. Its arsenal includes Doom, L.5 Doom, Aero, Ice 3, and a physical attack.
So casting Bum Rushing, breaking Fire Rods or Pearl Rods, Raging Io for Flare Star, or Throwing Flame Sabres will all do plenty of damage.
Alternatively, Vanish/Doom also works.
Blocking against Ice and Wind damage are both done with a Thunder Shield, Force Shield, Force Armor, Minerva, or a Cat Hood if you're bringing Relm.
Doom can be avoided fairly well with high M.Block or totally blocked with a Safety Bit or Memento Ring.
L.5 Doom can be avoided by simply not having a level divisible by 5.
3. I have no idea.