Kõiv attended school in Tartu after the second World War, graduating in the early 1950s with a degree in nuclear physics.
His first published work was a play called Küüni täitmine (Filling the Hay Barn) written as a collaboration between Kõiv (using his pseudonym Jaanus Andreus Nooremb) and Hando Runnel in 1978.
Just before the end of the decade, Kõiv began to publish works he had previously written for his own amusement under his own name.
In 1991 and 1993, he won the Tuglas short story award for Film and The Life of an Eternal Physicus, respectively.
He won the annual Estonian literary award again in 1995 for The Philosopher's Day and Return to Father.
Kõiv had released only 22 of the plays he had written and said that these comprise half of the dramatic literature he has created.
Kõiv's memoirs (the series Studia memoriae) consist largely of introspection, in sharp contrast to a typical biography.