Use the name of the card as it leaves the battlefield.

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.

Does not exile other cards of the same name that are on the battlefield. Just from the graveyard, hand, and library.

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.

The copies must be found if they are in publicly viewable zones. Finding copies while searching private zones is optional.

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.