Did the "Vatican State" even exist in 1861 as a valid nation-state? Didn't Mussolini and Cardinal Gasparri create the official division between Italy and the Vatican-city state when they signed the Lateran Treaty in 1929?
Before being known as the Vatican State, it was known as the Papal State, States of the Church, the Pontifical States, and Lo Stato della Chiesa.
It certainly did...sort of. What was left of the Papal States was a protectorate of the French at the time, whose troops kept King Victor Emmanuele from occupying the Patrimonium Petri. So the whole diplomatic recognition thing and that stuff about protecting confederates makes for an interesting, if not totally accurate, story. The Pope wasn't running the whole show at the time.