Q--Did the Eastern Orthodox Church and the Byzantine Empire ever use the same calendar?
A--My step-by-step reasoning: Eastern Orthodox Church used the Byzantine calendar from c. 691 to 1728. The Byzantine Empire used the Byzantine calendar from c. 988 to 1453.
So, the answer is yes.

Q--Did the leader of Heaven's Gate consider himself a prophet?
A--My step-by-step reasoning: The leader of Heaven's Gate was Marshall Applewhite. Marshall Applewhite said he was called to be a messenger of the divine.
So, the answer is yes.

Q--Did Leonardo da Vinci lack contemporary peers in his home city?
A--My step-by-step reasoning: Leonardo da Vinci was born in Anchiano, a town in the city of Florence. Da Vinci lived during the 15th and 16th century. Sandro Boticelli was a Florentine artist 15th and 16th century. Donatello was a Florentine artist during the 15th century.
So, the answer is no.

Q--Was P. G. Wodehouse's favorite book The Hunger Games?
A--
My step-by-step reasoning: P. G. Wodehouse died in 1975. The Hunger Games was published in 2008.
So, the answer is no.