Casting the card again due to rebound's delayed triggered ability is optional. If you choose not to cast the card, or if you can't because an effect prohibits it, the card will stay exiled. You won't get another chance to cast it on a future turn. If you do cast the card, it's put into its owner's graveyard as normal once it resolves.

If a spell with rebound that you cast from your hand doesn't resolve for any reason, including being countered, that spell won't resolve and none of its effects will happen, including rebound. The spell will be put into its owner's graveyard and you won't get to cast it again on your next turn.