Learn Hardcore German: Die Bibliothek gehört zur Universität.

zur is a contraction of zu + der, because Universität is feminine (die Universität). The preposition zu always takes the dative case, so the feminine dative article der merges with zu to form zur.

Which grammatical case is Universität in, and why?

It’s in the dative case. Any noun following zu must be in the dative.

Why does Bibliothek have the article die? Which case is it in?

Bibliothek is feminine and serves as the subject of the sentence. As a subject it takes the nominative case, whose feminine singular article is die.

Why is the verb gehört in the second position?

German main clauses follow the verb-second (V2) rule: the finite verb must occupy the second slot. After the subject Die Bibliothek, gehört comes next.

Could we say Die Bibliothek gehört der Universität without zu?

Yes. gehören can also take a direct dative object to express ownership (e.g. Das Buch gehört mir).
– Die Bibliothek gehört der Universität emphasizes that the university owns the library.
– Die Bibliothek gehört zur Universität emphasizes that the library is part of or affiliated with the university.

Are there any nuance differences between gehören zur Universität and gehören der Universität?

Yes.

gehören zur Universität = “to belong to (be part of) the university.”
gehören + dative = “to be owned by (someone),” focusing on possession.

Is it possible to start the sentence with Zur Universität? What happens to word order?

Yes. You can front the prepositional phrase:
“Zur Universität gehört die Bibliothek.”
You still apply the V2 rule, so gehört remains the second element and the subject die Bibliothek follows.