Breakdown of Heidän pitää nähdä itse paavi ennen valtiovierailua.
he
they
nähdä
to see
itse
self, oneself
ennen
before
Questions & Answers about Heidän pitää nähdä itse paavi ennen valtiovierailua.
Why is itse placed before paavi, and why didn't its ending change?
This is the intensifying use of itse, meaning "the pope himself" or "the pope in person". When used directly in front of a noun to emphasize it, itse is an uninflected word. This means it stays exactly as itse and never changes its form, regardless of what grammatical case the noun after it is in.
Since paavi is the object of the sentence, why isn't it in the genitive or partitive case?
Because it is the object of a necessity sentence. When a sentence uses an obligation verb like pitää (must/have to), a total object that would normally take a genitive -n ending appears in the basic nominative case instead. Therefore, paavi looks just like its dictionary form here.
Why is the subject Heidän in the genitive case instead of the nominative He?
Verbs of obligation like pitää and täytyy require a specific "necessity sentence" structure, where the person who has to do the action is put into the genitive case. You can think of it as literally meaning "Of them, it is required to see...".
Why does valtiovierailua end in the partitive -a?
The word ennen (before) functions as a preposition here, and it requires the noun that immediately follows it to be in the partitive case.