You pay all costs and follow all timing rules for cards 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 and only if you have an available land play remaining.

If another player gains control of Lightning, you won't be able to play any of the exiled cards, even if you later regain control of Lightning. Similarly, if Lightning leaves the battlefield and later returns, you won't be able to play the cards that were exiled the first time it was on the battlefield since it's a new object.