Eradicate

• WWizards of the Coast2004-10-04
  Official Ruling
  Use the name of the card as it leaves the battlefield.
• WWizards of the Coast2004-10-04
  Official Ruling
  Does not exile other cards of the same name that are on the battlefield. Just from the graveyard, hand, and library.
• WWizards of the Coast2004-10-04
  Official Ruling
  If a copy card is targeted by this effect, you get to look for another copy of the card it is copying. This is because a copy card actually takes on the name and initial characteristics of what it copies.
• WWizards of the Coast2005-02-01
  Official Ruling
  The copies must be found if they are in publicly viewable zones. Finding copies while searching private zones is optional.
• WWizards of the Coast2005-02-01
  Official Ruling
  If you manage to turn a land (such as a Forest or Stalking Stones) into a creature, you can indeed use this effect on that basic land type. Eradicate’s exile effect only looks for cards by name, not type.
• WWizards of the Coast2012-05-01
  Official Ruling
  A face-down creature has no name, so no card can possibly share a name with it (not even other cards with no name). If you Eradicate a face-down creature you will still search its controller’s library, but you won’t be able to exile any cards from that library.

Rulings via Scryfall