Yes, it was a real opera: "Cavalleria Rusticana" ("Rustic Chivalry") by Mascagni. I also didn't like the way Tony's aria (the first one in the opera, "Siciliana," was sung). But it's a great opera when done right--short, ugly, but, IMO, with the highest concentration of great music in a short (~80 minutes) running time.

What was FFC thinking? It was a wise choice for GFIII. I saw it as typifying the whole Sicilian "thing" that Michael was involved with all his life:

It's about infidelity, cuckolding and (relevant to GFIII) violent revenge. The lead character, Turiddu (played by Anthony) has an affair with the wife of Alfio, in the process jilting Santuzza, whose honor he stole and then abandoned. Santuzza, heartbroken and wanting revenge, tells Alfio that Turiddu is cuckolding him with his wife. So, Turrido and Alfio have to fight to the death to achieve revenge and restore honor. Note that FFC showed a confrontation between Turiddu and Alfio in which Turridu bit his ear--the traditional Sicilian challenge to a fight to the death over honor. Notice how Vincent grins when it happened? Recall that he bit Zasa's ear early in the movie--for the same purpose.