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If I put a double-faced card onto the battlefield with Manifest (Put it onto the battlefield face down as a 2/2 creature. Turn it face up any time for its mana cost if it’s a creature card), what happens? Does it enter the battlefield transformed, is it a 2/2 with no abilities, or is it not even possible to be manifested?

This article explains that double-faced cards can be manifested and enter as 2/2's with no abilities. The actual implementation of this is that you either have the card in an opaque sleeve or you have a checklist card (which is necessary for the double-faced card to not be identifiable in your hand and library) and either of these allows it to be put face-down on the battlefield as a state distinct from transformed.

It should be noted that double-faced cards can't be turned down (such as by Ixidron. The relevant rules are the following:

711.9. If an effect allows a player to cast a double-faced card as a face-down creature spell, or if a double-faced card enters the
battlefield face down, it will have the characteristics given to it by
the rule or effect that caused it to be face down. That card remains
hidden, using either a face-down checklist card or opaque sleeves. See
rule 707, “Face-Down Spells and Permanents.”

711.9a While face down, a double-faced permanent can’t transform. If it is turned face up, it will have its front face up.

711.10. Double-faced permanents can’t be turned face down. If a spell or ability tries to turn a double-faced permanent face down, nothing
happens.