You pay all costs and follow all timing rules for cards played this way. For example, if the exiled card is a land card, you may play it only during your main phase while the stack is empty.

Tersa Lightshatter’s last ability checks your graveyard at the moment it would trigger to see if you have seven or more cards in your graveyard. If you don’t, the ability won’t trigger at all. If it does trigger, the ability will check again as it tries to resolve. If you don’t have seven or more cards in your graveyard at that time, the ability won’t resolve and none of its effects will happen.