Eye of Doom

• WWizards of the Coast2013-10-17
  Official Ruling
  The active player chooses a permanent first, followed by each other player in turn order. Each player will know what previous players have chosen when making their own choice. Then the doom counters are put on all the chosen permanents simultaneously.
• WWizards of the Coast2013-10-17
  Official Ruling
  None of the permanents are targeted. A player can choose a permanent with protection from artifacts, for example.
• WWizards of the Coast2013-10-17
  Official Ruling
  If a permanent with a doom counter isn’t destroyed by Eye of Doom’s last ability (perhaps because it has indestructible or it regenerated), the doom counter will remain on that permanent.

Rulings via Scryfall