Short answers, pulled from the story.
The actual cathedral of the Holy See is the Archbasilica of Saint John Lateran in Rome. This distinction defines the very nature of the office, which traces its origins to the first century when Saint Peter and Saint Paul established the Diocese of Rome.
The Holy See was granted sovereignty by the territorial Donation of Pepin in 756 by King Pepin of the Franks. From 756 to 1870, the Papal States held extensive territory and armed forces.
Cardinal Pietro Parolin is the incumbent Cardinal Secretary of State of the Holy See. Parolin was named in his role by Pope Francis on the 31st of August 2013.
The Holy See maintains diplomatic relations with 180 states. It is the only European subject of international law that has diplomatic relations with the government of the Republic of China as representing China, rather than the government of the People's Republic of China.
The Pontifical Swiss Guard was founded by Pope Julius II in January 1506 as the personal bodyguards of the Pope. It continues to fulfill that function and is listed in the Annuario Pontificio under Holy See, not under State of Vatican City.
The Holy See is the universal government of the Catholic Church and operates from the Vatican City State. The State of the Vatican City was created by the Lateran Treaty in 1929 to ensure the absolute and visible independence of the Holy See and to guarantee to it indisputable sovereignty in international affairs.