When the last time counter is removed from the exiled card, it's cast as a completely new spell. Modes and targets are chosen again. If the spell has any additional costs, they must be paid again. If they can't be, the spell can't be cast and stays exiled.
If the spell targeted by Delay was cast with flashback, Delay's effect will exile it, not the flashback effect. The card will get time counters and gain suspend (if it didn't already have suspend).