The aria at the beginning (I haven't seen the whole episode) is "Ebben? Ne andrò lontana" from Act I of the Italian opera La Wally by Alfredo Catalani.
Vissi d'Arte
Nessun dorma
His most famous aria is his Air on the G string. While Bach's Air on the G String (a piece from his Orchestral Suite No. 3 in D major, transposed to the key of G) is very popular, it probably wouldn't qualify as an aria.
If you mean the version that is used on the Jean Paul Gaultier 'Le Male' and 'Classique' adverts - This is a famous track from the Vincenzo Bellini opera Norma (Casta Diva). In my opinion, Maria Callas delivers the best version.
Calaf
Aria does it because that's her character, now if you ask why Lucy hale does it then I think it's because she was the it's famous out of the four.
The most famous aria from Verdi's opera "Aida" is "Celeste Aida." Sung by the character Radamès, this aria expresses his deep longing and love for Aida, an Ethiopian princess. Its beautiful melody and emotional depth have made it a highlight in operatic performances and a favorite among audiences. The aria showcases both the lyrical and dramatic qualities characteristic of Verdi's compositions.
The famous Habanera - L'amour est un oiseau rebelle.
There is an exceptionally famous aria that Figaro sings in act 1 ' Largo al factotum della città' - 'Make way for the factotum of the city'
Flower Duet (Lakme: Viens, Mallika...Dome Epais)
aria aria