In the Broadway musical The Phantom Of The Opera, why does Christine die? Why does the Phantom let Christine and Raoul go in his lair? Does Meg...

That's a lot of questions--I'll try my best to help!

First of all, Christine doesn't die in either the musical or the book.  (She does meet her end in Love Never Dies, Andrew Lloyd Webber's sequel to Phantom that has no basis in Leroux's original work, but that musical was widely panned.)  I don't believe the musical explains what happens to Christine and Raoul after they leave the Phantom's lair, but in the book, they leave Paris.  Leroux presumes they've stolen away and gotten married.

The Phantom ultimately lets Christine and Raoul leave his lair because Christine shows him compassion, the first time he's been on the receiving end of any kindness at all.  She also kisses him, which almost certainly contributes to his more charitable mood, but in the end it's Christine's kindness that enables him to find that scrap of kindness in himself.

As for the question of the Giry women and the Phantom... given that Meg leads the angry mob to search for the Phantom after Christine's kidnapping, I find it a little hard to believe she would have secretly fallen in love with him, though it's worth mentioning that the comic strip Little Meg envisions a world in which a considerably younger Meg has an almost Calvin-and-Hobbes-like rapport with the Phantom.  You raise a good point regarding Mme. Giry; her history with the Phantom is an invention of the musical and wasn't in the original book, so that would seem to lay some groundwork for future romance.  Ultimately, though, this story is very much in the Gothic tradition, and one hallmark of Gothic stories is the innocent, virtuous heroine to whom many unimaginable, horrible things happen.  Christine fits that mold to a T--she's naïve enough to believe in the Angel of Music, she's very young, and she's presented as the virginal counterpart to Carlotta,  to say nothing of her manipulation and kidnapping at the hands of the Phantom.  Mme. Giry, on the other hand, is considerably older and also a mother.  In many ways, the Phantom is preying on Christine, and she's simply far more attractive prey than Mme. Giry.