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

If Headliner Scarlett leaves the battlefield, the effect allowing you to look at and play the exiled card continues to apply for the rest of the turn.

Once the turn ends, you can't look at or play the card exiled with Headliner Scarlett that turn. It remains face down in exile, and no one can look at it.