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 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.

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.