The exiled card is played using the normal timing rules for its card type, as well as any other applicable restrictions such as “Cast [this card] only during combat.” For example, you can’t play the card during an opponent’s turn unless it’s an instant or has flash. Similarly, if the exiled card is a land, you can’t play it if you’ve already played a land that turn. If it’s a nonland card, you’ll have to pay its mana cost. The only thing that’s different is you’re playing it from the Exile zone.

If you do not play the card before your next upkeep, it remains exiled; you just lose the ability to play it.

This is not considered to be drawing a card.

If the Bottle leaves the battlefield or your control, the cards remain waiting until played or until the beginning of your next upkeep. The card is in the “Exile” zone.

The card is not part of your hand in any way. You can’t be forced to discard it due to a discard from hand effect, and you can’t discard it to pay a cost.

To “play a card” is to either cast a spell or to put a land onto the battlefield using the main phase special action.