A pope goes through the same canonization process as any candidate for sainthood. He gets no special consideration. Information about the man is gathered by the postulator of the cause. He organizes it, investigates any miracles, and presents the case to the Sacred Congregation for the Causes of Saints. They then make a recommendation to the current pope to either canonize or not canonize the candidate.
Pope St. Martin is a saint but not a patron saint.
Saint Patrick was never a pope.
The pope, upon election, chooses his regnal name. He is not required to do so and may keep his birth name if he so chooses.
Yes, Saint Peter was the pope at the time.
Pope St. Nicholas I is not a patron saint.
Pope Saint Sixtus I - April 6 Pope Saint Sixtus II - August 7 Pope Saint Sixtus III - March 28
No, Pope Saint Peter was the first pope.
There was a Pope Leo XIII but he has not been canonized as a saint.
He would have been appointed to the position by the pope. In the case of Saint Nicholas, that was probably Pope Saint Sylvester I.
no.
Pope Saint Pius I was the tenth pope.
Pope Saint Pius X was pope in 1912.