1/ What is cesaropapism ?
It is the trend/political will or aim to gather both temporal and spiritual powers within one head (cesar : political, papaism <-- pope : spiritual) to use them to reinforce one another.
2/ What was the language spoken at the court of Catherine II of Russia ? Bonus : what was Catherine II's origin ?
There was a trick here : she was coming from a German-speaking area (then) but the language spoken at the court was the universal language of the time : French. Many nobles even pretended not to know (or to forget) Russian !
3/ What is an "old-believer" ?
The Russian orthodox who wanted to keep the same traditions as before the reforms of Peter the Great (wear long beards for instance).
4/ Janissaries were made up of Christian children taken as taxes when still young. Spahis were both similar and different in their (at least initial) recruitment. How ?
Spahis were mostly (at least at the origin) Christians as well. But unlike janissaries they were free men, often renegades from Austria, Russia, Poland, Balkans or simple mercenaries recruited in Christian Ottoman-controlled areas.
5/ Joseph II of Austria did not bother to get the crown of St Etienne. What did it refer to ?
The crown of St Etienne is the crown of Hungary. For Austria was not one country but the same king having different crowns (with different administrations, taxes,... as well). Joseph II wanted to unite his holdings and give them a certian unity therefore he dismissed the crowns of Hungary and Bohemia. But instead of uniting the people it gave them more reasons to complain as the people of these areas felt despised, second-ranked people.
6/ What was the title (prince, Lord,....) of Haile Selassie of Ethiopia when Italy invaded it before WWII ? I want the official title, not a translation of its meaning.
His title was simply the "Negus".
7/ Taiwan was know before as "Formosa". What language did that name come from ? Bonus : what did it mean ?
It is coming from Portuguese. It means the beautiful.
8/ What battle (in 642) between Arabs and Sassanids saw the fall of the latter ?
Nehavend. Ctesiphon was also a major battle a few years before but Nehavend really saw the crumbling of the Sassanids.
9/ Cook proved in his 1770 trip that New Guinea were separated by a straight. Someone had discovered it before but no other trip before Cook had been really able to prove it. Who was he ?
OK I made a mistake in typing the question. I wanted the name of the straight (straight of Torres). So I will accept as answers Torres or Tasman.
10/ In the last quarter of the XIXth century, a war started between Peru, Chili and Bolivia for the coastal area. Who lost and who got the disputed area ? Bonus : what was the name of that war ?
Bolivia lost all access to the sea. The war is known as war of the Pacific or war of the saltpeter as the area they fought for was extremely rich in saltpeter and guano (fertilizer made up of birds' rejections). XIII mentionned "Chaco war". I must say I never heard that name before but it might be true. But I did not count it as correct. Now if you can find evidence, I will gladly change your score.
Scores (only 3

)
XIII : 5.5
Dragonlord : 3.5
Redtooth : 3
So it is XIII's turn once again (or whomever he lets do it)