I am going to answer this part by part from your subquestions rather than one chronological narrative, though if requested, I can always do that as well, and if some specific follow-up question would arise.

The relations between Serbia and Austro-Hungarian Empire was notoriously hostile, specially so after unilateral annexation of Bosnia ( which per Berlin treaty was under its occupation, but not de facto part of its Empire ), without notifying either Serbia or the Ottomans, and only notifying Russia a week prior to it, but then diplomatically outmaneuvering, and consequently damaging those relations as well. They eventually paid the Ottomans off, but all the other hostilities and resentment remained. After the Serbian victory in the second Balkan war, the relations took another hit, with Austria securing their Balkan interests.

Gavrilo Princip was a Bosnian Serb, seeking independence for Yugoslav nations from Austria, so the term might not be that accurate. Overall, the Bosnian Youth movements ( later called Mlada Bosna ) were diverse, student-based from catholic Croats, muslim Bosnians and orthodox Serbs, among others.

I am incredibly confused as to whether or not Serbia was part of Austria-Hungary

Serbia was autonomous since 1830, independent since 1867-76 ( depending on conditions ), internationally 1878, proclaimed kingdom in 1882.

and if not why and how the Austria-Hungarian crown prince could have been killed in Serbia if it was not part of the empire

Namely, it was not in Serbia, but Bosnia, Sarajevo, where the Archduke was killed, was occupied by Austria-Hungary since the Berlin treaty, and part of of the Empire since 1908.