You pay all costs and follow all normal timing rules for cards played this way. For example, if one of the exiled cards is a land card, you may play it only during your main phase while the stack is empty.

Only cards still exiled with Apple of Eden’s ability will be returned to their owner’s hand at the beginning of the next end step. Any cards that left exile (probably because they were played) won’t be returned to their owner’s hand, even if they somehow returned to exile face down.

Only cards in the target opponent’s hand when Apple of Eden’s ability resolves will be exiled face down. Any cards they draw later in the turn, including cards drawn because of Apple of Eden’s first delayed triggered ability, will simply go to their hand. You won’t see those cards, they won’t be exiled face down, and you won’t be able to play them.

If you leave the game before the second delayed trigger resolves, the cards remain exiled face down indefinitely. No player may look at them.