Nightveil Specter’s last ability applies to cards exiled with that specific Nightveil Specter, not any other creature named Nightveil Specter. You should keep cards exiled by different Nightveil Specters separate.

Playing a card exiled with Nightveil Specter follows all the normal rules for playing that card. You must pay its costs, and you must follow all timing restrictions, for example.

The card is exiled face up. All players may look at it.