Treat Body Double as though it were the chosen card entering the battlefield. Any "As this card enters the battlefield," "This card enters the battlefield with," and "When this card enters the battlefield" abilities of the chosen card will work.

You don't have to choose a card to copy. If you don't, Body Double enters the battlefield as a 0/0 creature and is probably put into your graveyard immediately, unless something is increasing its toughness to keep it alive.

If Body Double copies a Vesuvan Shapeshifter which then copies another creature and then is somehow turned face down, the Body Double is now a face-down Vesuvan Shapeshifter and may be turned face up via its Morph ability.