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In the very first scene, Fleck calls Mme. Giry, not the proper French "Madame," but "Madam." This could be just her accent, but considering that in the slang of the time, a "madam" was a woman who owned a brothel...
Maybe it was a subtle hint to how they managed to raise the money for the sideshow?
Remember the name of the ship that Christine comes to Coney Island on? The Persephone. Now, why does this seem strangely fitting? Time for a quick mythology lesson: Persephone was the daughter of Demeter, the goddess of the harvest. One day, she was kidnapped by Hades, who had become obsessed with her beauty and desired to make her his bride. Demeter was so distraught by the loss of her daughter that she made the earth grow cold and the plants die. Eventually, the gods made Hades return Persephone, but not before he got her to eat six pomegranate seeds—dooming her to have to return to the underworld for six months of each year, during which her lonely mother causes the world to be cold again. Now let's look at Love Never Dies. Phantasma is the underworld (full of darkness and freaks/monsters), the Phantom is Hades, and Christine is Persephone. In Phantom of the Opera, Christine was kidnapped by the Phantom, who wanted to make her his bride, and was obsessed with her beauty. This is still the case in Love Never Dies. The aria that the Phantom wants Christine to sing are the pomegranate seeds—if Christine "eats them" (sings), she has to stay. She does "eat," but the Phantom's actions removed the warmth from the world (think the beginning, when we see that everything was ruined by the actions in the musical), and he doesn't get to keep her entirely, but he gets Gustave, a "half year" of beauty.
Considering that the Phantom's teachings made Christine an internationally celebrated singer, one should consider that Meg has a crush on him in the hopes that he'll make her as famous.
Gustave's conception. The book has Christine actively seeking Erik out the night before her wedding!!! Even worse, he in a fit of morning-after guilt, abandons Christine without so much as a word of explanation, leaving her alone and ultimately pregnant. Considering what might have happened to her if anyone found out about this — Fantine in Les Miserables, anyone? — that was highly irresponsible. (Making the rape interpretation of "Music of the Night" canon would have been somewhat logical, but then she probably would have given birth before the end of the first show.)
Christine to the Phantom: "Forgive me, I beg you, if you can/I've brought you nothing but woe." WHAT? To refresh your memory: she is talking to the man who stalked her, deceived and manipulated her, terrorized and murdered her colleagues, kidnapped her, and nearly forced her into a Scarpia Ultimatum marriage. Then he slept with her, only to run away like a thief in the night leaving her abandoned and alone, decided ten years later that he still wasn't ready to let her go, took advantage of her family's precarious financial situation to drag her across the ocean so he could mess in her life again, and behaved in a frankly sinister manner to her son before learning that it was his son, too. And she's apologizing to him? Talk about Karma Houdini!
Even more glaring is the issue of how Chirstine can even be so sure whose kid Gustave even is. Paternity tests weren't available back then, so how can she know? It implies she and Raoul hadn't consummated their marriage, in which case he'd logically suspect infidelity. But if such were the case, why is Raoul so surprised when the Phantom suggests Gustave might not be his? Makes too little sense.
The Phantom's logic for concluding that Gustave is his son is pretty shaky too: the only evidence is Gustave's musical ability and his fascination with the macabre. The former could have come from his mother and maternal grandfather, and the latter is hardly extraordinary in a ten year old boy.
Maybe the fact that Gustave has no younger siblings plays into Christine and the Phantom's conviction that Raoul isn't Gustave's father. They might figure that, since there are no children of a ten-year marriage, Raoul is sterile.
It can be very difficult to have children sometimes. Just look at how many disasters happened because royal couples had trouble producing a child.
The Phantom's wrong: Gustave can't be his son! The first musical takes place in 1881, and the second in 1907. That's not ten years. The only person who says it's been ten years is the Phantom, who also claims to have been counting the days "till I hear you sing once more." He's probably been counting the tenth year for a long time now in his delusion that she'll come back to him. Gustave is ten, which just so happens to fit the Phantom's delusion. And since Christine just submits whenever the Phantom is around, she played along, even though there is no possible way that it could be!
I'm not entirely sure if this is the right trope to invoke, but Word of God has stated that this story takes place ten years after the events of Phantom, despite the chronological errors made. As it states on the main page, Writers Cannot Do Math. So from this troper's perspective, the Phantom can be the father. The likelihood of that, however, has already been debated above.
Because we can't probably have Raoul The Disgusting Alcoholic polluting the gene pool! No way!
Why would a burlesque dancer and a world-renowned opera diva be performing in the same show?
Christine and the Phantom sleeping with each other the night before Christine married Raoul has some logistical issues in the first place. Presumably it happened before the Phantom fled to America, but it seemed implied that he got out of France right away? So theoretically at the time Gustav was conceived, Christine would have been in Paris and the Phantom would have been in New York.