A card with two faces always enters play as the front face. Whenever it is not in play, it is considered to be the front face. When a game effect instructs you to flip it, it is transformed into the card on its other face, and loses all abilities and attributes on its other face.

Whether a card is rezzed/derezzed is independent from whether it is flipped/unflipped. So if Caterpillar is first rezzed, then flipped to Monarch, then derezzed, if it is later rezzed again it would be as Monarch, not as Caterpillar. This is because derezzing a card does not cause it to leave play.

For paper play, flip cards can be handled by using opaque sleeves, and when flipping them, removing them from their sleeve, turning them over and reinserting them. (Alternatively, substitute cards can be used.) If a player ever finds a flipped card in their hand, deck or discard pile (indicated by a black title box), they know that a mistake has taken place, and can correct it by flipping the card over.