Chancellery language. Medieval Lithuania, much like, I imagine, all other states at the time, had no designated "official" language. It's just that since Chancery Slavonic was used by nearby Ruthenian states whom Lithuania conquered, and thus it was more easy for the Lithuanians to use that language for official text rather than import Western scholars to write texts in Latin.
It was a choice of pragmatism, not any ethnic or linguistic allegiance. Had Lithuania expanded to the West instead of East, Latin would have been their chancellery language for the same reasons.
Not really.
The Lithuanian term for the ruler of Medieval Lithuania, "didysis kunigas" (later evolved to "didysis kunigaikštis", as kunigas began to mean priest), most accurately translates to something like "great duke". Or probably even "great king", as kunigas is a linguistic borrowing of the German König.
I mean, in my opinion, calling Medieval Lithuania "Grand Duchy of Lithuania" is an anachronism, but so is "Grand Principality of Lithuania", for the same reason.