The Phantom of the Opera/WMG

The Phantom being handsome is intentional.
He's in the circus as a freak because he was genuinely bad-looking enough at the time for that to work--barely (and the fact that he was so starved/scrawny was necessary for it to work). As he grew, his face got better, but he was so conditioned to believe otherwise that everyone else believed it. When Christine takes off his mask, he yells at her while...covering his face, so she believes it. When he's on stage, no one could see his face at all since he was facing sideways--they gasped at the dramatic reveal and assumed that the person who had kidnapped their beautiful young star must have been ugly, since that's how it works.
 * Except he was facing stage left during the unmasking, meaning the "ugly" right half of his face was facing the audience.

The Phantom is Meg's father.
Not quite sure about the timeline matching up, but come on! There's some sort of subtext between Madame Giry and the Phantom- there has to be some reason she puts up with his murdering psychopathic ways- and I don't believe Meg's father is ever mentioned.
 * Seconded - and I thought I was the only one who ended up Shipping the Phantom with Madame Giry before the novel was over.
 * If so, then the plot of Love Never Dies just got a lot more Oedipal - not only a spurned adorer, but a rejected daughter!
 * Nope, Madame Giry says her husband's name was Jules, and he died years ago...
 * Just because he was her husband doesn't mean he was Meg's father...

The Phantom really is a phantom, and he may not even realize it.
Specifically, he's a Noble Demon who's honestly trying to be an angel -- an Angel of Music! But he fell back to his evil tendencies when he became desperate for Christine. But when he received just a bit of her compassion, just a bit of her love at the end, he felt restored, enough to be willing to let her and Raoul go. He even appears to vanish at the end, like a ghost finally put at peace!

Andrew Lloyd Webber is either the Phantom himself or Gustave.
Just think about it for a moment. The Phantom (and, it is implied, Gustave) is a musical genius and has a passion for darkness. Andrew Lloyd Webber is most definately a musical genius. As for the darkness:
 * "Macavity"
 * "Damned For All Time", as well as the whole second act of JC Superstar
 * "Sunset Boulevard"
 * The beginning of "Evita", not to mention that Peron's speeches sound like the Phantom ranting to this troper.
 * Also, "Theme and Variation" sounds like something the Phantom would program that four-armed techno gorilla to play on the organ in "Love Never Dies."
 * In support of this WMG, the fact that Lloyd Webber and Sarah Brightman (who originated the role of Christine) were married at the time.
 * The role was specifically written for Brightman, after all. It wouldn't be surprising if the Phantom was supposed to represent Webber, especially when you know how the two met each other.

The Phantom turns up years later as Sander Cohen

 * No doubt both of them have deformed appearances and wear a mask (though Cohen's is a rabbit mask), and both of them are even mad artists. Could perhaps be that Sander Cohen's past (as officially mentioned) was a fabrication if such were the case.

Love Never Dies is a Stealth Parody of bad fanfiction.
Several aspects of the musical--the shaky chronology, the Phantom and Christine drifting into Mary Sue territory, the extraordinary Character Derailment (especially for Raoul),, Meg being attracted to the Phantom, enormous amounts of Wangst all around--appear so often in Phantom fanfiction as to be cliches. Since there's no way a bunch of renowned, respected artists would present, in all seriousness, a story that could have been written by a fifteen-year-old under the alias of "The Future Mrs. Gerard Butler," they must have intended it as a satire....right?
 * Frederick Forsythe, the author of the source book, claimed that he wrote Phantom of Manhattan as a Take That to gothic literature, so the music could be an adaptation of a parody. However, as ALW commissioned Forsythe to write that book for future sequel purposes, it would be in pretty bad taste to screw over a commissioner like that.
 * So, Forsythe could have been commissioned to write a serious sequel but intentionally secretly made it a Stealth Parody without ALW catching on. This opens theories of Forsythe as a Christine/Raoul shipper (showing the absurdities of the pro-bad boy side of Gothic fandoms) and of ALW as a Christine/Erik shipper.

The original novel was the Narrator's attempt Gone Horribly Right to make amends to the Phantom, whom he canonically saw as a Woobie Destroyer of Worlds.
Since the Phantom was separated from the woman he loved, the Narrator intentionally portrayed him in a light that made him sympathetic and irresistable to legions of fangirls the world over for decades (which will no doubt turn to centuries).

The real reason the Phantom let Christine go...

 * So he gives her the ultimatum... Say you love me, I'll let Raoul go and we live together forever. Say you love Raoul, and I'll kill him. She decides to come up and the Phantom. He stops, and lets them both go. Why? Christine's a terrible kisser, and the Phantom is just really shallow.
 * I thought he just discovered that he was gay that very second.