The "legend rule" checks only for permanents with exactly the same name. Elbrus, the Binding Blade and Withengar Unbound can be on the battlefield at the same time without either going to its owner's graveyard.
Elbrus will transform even if you are unable to unattach it (most likely because the equipped creature died).
Withengar's triggered ability will trigger no matter how a player loses the game: due to a state-based action (as a result of having a life total of 0 or less, trying to draw a card from an empty library, or having ten poison counters), a spell or ability that says that player loses the game, a concession, or a game loss awarded by a judge.
In a multiplayer game using the limited range of influence option (such as a Grand Melee game), if a spell or ability says that you win the game, it instead causes all of your opponents within your range of influence to lose the game. This is another way by which Withengar's ability can trigger.
For more information on double-faced cards, see the Shadows over Innistrad mechanics article (http://magic.wizards.com/en/articles/archive/feature/shadows-over-innistrad-mechanics).