You don't have to choose a card to copy. If you don't, Body Double enters 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.

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