You must follow any applicable timing permissions and restrictions if you cast the exiled card. This means that if Vivien leaves the battlefield, you can't cast the creature spell as though it had flash unless another effect allows you to do so.

Once you cast the creature card, it leaves exile. You can't cast it again if that card becomes exiled again.

While resolving Vivien's last ability, you must exile one of the three cards, even if none of them are creature cards. You may look at the exiled card at any time, even if it's not a creature card.