Michael and dbb both had majority right, although I didn't figure up their exact grade. I hope this was enjoyable and thought provoking.
1) Prior to John Paul I, when and who was the last pope to be the first to take a particular name?
Landus, who reigned from 913-914 AD
2) Pope Benedict XVI is the 263rd legitimate pope, how many unique names have been used by popes?
(With the caveat about Stephan II who was elected but died before being installed, and Benedict IX who served three times, making Benedict XVI the 265th successor of Peter;) there have been 89 unique names used by the popes.
3) After St. Peter, who is the longest reigning pope?
Yep, Pius IX, 31 years, 7 months, and 22 days
4) Which century has had the fewest popes? (Not counting the present one, that's cheating!)
The Nineteenth Century, with six, had the fewest for a complete 100 year period. (Certainly Pius IX’s long reign helped, as did Pius VII and Leo XIII). The First Century had five.
5) Which century has had the most popes?
(This one took some time to figure!) The winner is the Tenth Century with 23 in a hundred year period! The Seventh and Ninth Centuries tied with 21. (I counted the aforementioned Benedict IX as one, therefore 20 in the Eleventh.)
6) Which name has gone the longest between the first and second iterations?
Pius, at thirteen hundred years: Pius I (142-157) to Pius II (1458-1464); Paul was a ‘mere’ 700 years.
7) Before John Paul II, who was the last non-Italian pope?
Adrian VI who reigned from 1522-1523
8) How many popes were there during the Avignon Captivity?
Seven (Pope Clement V in 1305 to Gregory XI in 1376; however two Anti-Popes were elected in Avignon: Clement VII and Benedict XIII, who was expelled from Avignon in 1403)
9) What is the name of the first pope, alphabetically?
10) What is the name of the last pope, alphabetically?
In answer to the Bonus Question posed by Michael in the combox (you know, this is a death knell to any agenda, hence I’m glad to do it!): “If elected Pope, what name would you choose?” Probably Eugenius V, the previous four were generally strong popes who helped advance the development of the papacy. (Plus, it would be in honor of my grandfather.)