You pay all costs and follow all normal timing rules for cards played with the permission granted by Strongbox Raider's ability. For example, if the exiled card is a land card, you may play it only during your main phase while the stack is empty.

Raid abilities care only that you attacked with a creature. It doesn't matter how many creatures you attacked with or which player, planeswalker, or battle those creatures attacked.

Raid abilities evaluate the entire turn to see if you attacked with a creature. That creature doesn't have to still be on the battlefield. Similarly, the player, planeswalker, or battle it attacked doesn't have to still be in the game or on the battlefield.