L'ultimo esame in settimana!!

È l'ultimo esame, della settimana che chiude il trimestre autunnale! (Why is it not used as "autunno"?)
Buona fortuna a te!

(Is there a slang word for "final exams" in Italian? In English we call it "finals.")

Eseme Finale Settimana!! --> La settimana degli esami di fine semestre

"esami finali" it isn't really said, better too "esami di fine corso"/of the end of the course. 

About this : end course => of the end of the course, like in Italian it would be better write => di fine corso/della fine del corso Without preposition it sounds like headlines : Settimana fine esami. It is correct but it is a specific register. 

È finale settimana per il semestre autunnale! 

E' l'ultima settimana del semestre autunnale.

(finale is the last, and it has a meaning of a longer end, more conclusive)

(Why is it not used as "autunno"?

"autunno" is the noun so it would become "del semestre d'autunno"

"autunnale" is the adjective.

You can use them according your expressive aims)

(Is there a slang word for "final exams" in Italian? In English we call it "finals.")

(We don't have exam sessions similar to yours, neither in high school -last exam is named "esame di maturità"-, nor at university, so I don't find an equivalent expression :-) )

Good luck !