There's a devil like person, and a lot of flying creatures sort of like bats, which is why they had to leave the theatre, but what Opera were they watching?
The opera is called Mefistofele, which is a Faustian opera (i.e. dude makes a deal with the Devil). Wikipedia states:
Batman Begins depicts the opera being performed onstage, using an excerpt of Rampiamo, rampiamo, che il tempo ci gabba (Chorus of Warlocks and Witches from Act 2, scene 2) from the 1973 EMI (see "Recordings" above). During the scene, performers dressed as bat-like monsters frighten young Bruce Wayne, who asks to leave.
In the opera, the title character is the Devil. Mefistofele makes a bargain with the Forces of Good that he can corrupt Faust, and spends the opera trying to do so. He does this by making deals with Faust, to fulfill Faust's wishes. Faust pines for a woman, Mefistofele enables him to seduce the woman, but then the woman realizes Faust's relationship with Mefistofele and she seeks God's forgiveness. This sort of thing happens a few times. Eventually Faust realizes that neither the real nor the illusionary pleasures that Mefistofele offered were the perfect bliss Faust sought. He grabs his Bible, seeks God's forgiveness, and receives it. Mefistovele is then sent back to Hell, having failed to turn Faust.
Essentially this mirrors Batman's arc in the film. Faust is Bruce, and Mefistofele is Ra's. Batman is then God, I think?
Yes, it's Faust but the demon's name is Mephistopheles, shortened to Mephisto. Faust is based on Goethe's Faust which has two parts. The second part eventually ends with Faust's soul saved by learning to care about others and finding that is the ultimate experience he was seeking and he ascends to Heaven despite his deal with Mephisto.