Cash Cow Franchise: The London and New York productions have been up and running since 1986 and 1988 respectively, and have twice out-grossed all of the Transformers movies combined. Tours and foreign productions are similarly popular, and a lot of merchandise follows in their wake. In Vegas, there was a special condensed 95-minute version that retained most of the songs.
Dawson Casting: While most actresses to play Christine in the stage show are in their early twenties, a handful have been in their thirties or even nearing forty. The book establishes that both Christine and Raoul are in their early twenties.
Development Hell: The film version of the musical was announced in 1989 but didn't start production until 2002, and wasn't released until 2004.
Fake Nationality: Everyone in nearly every single production, given that the story is set in Paris (and Christine is Swedish (though this has of course been averted in the Swedish production), while Carlotta and Piangi are Italian).
Playing Against Type: Michael Crawford was a well-established light comedy/musical actor in the U.K. when he originated the role of the Phantom in the original London staging. Even he was surprised at the casting. He has been quoted saying that when Andrew Lloyd Webber told him he wanted him for Phantom, he thought he meant for the part of Raoul.
A similar thing occurred with Peter Jöback, who took on the role in London in 2012 (and later played it on Broadway and in his native Sweden). Jöback had previously originated the roles of Robert in Kristina from Duvemåla and Michael in The Witches of Eastwick, played roles such as Chris in Miss Saigon, and provided the Swedish voice of Aladdin (opposite, coincidentally, the original Swedish Phantom, Mikael Samuelsson, as Jafar). Jöback sought the role of the Phantom but due to his voice (he's a tenor, like Crawford), previous credits and somewhat boyish looks was offered Raoul instead. He declined, feeling he had played that kind of part enough times and wanted to do something new. Andrew Lloyd Webber then contacted him and offered him Phantom.
Stunt Casting: Averted in most cases, as the show never depended on big name stars. An exception in Paul Stanley of KISS fame being cast as The Phantom in Toronto.
Another exception would be the Swedish 2016 revival, which more or less happened because Peter Jöback is a big star in Sweden, and him playing the role in London and New York brought enough interest in the show to bring it back (shows generally don't play for more than a few years in Sweden due to the population size).
At the end of the show, as Christine leaves the Phantom, he picks up her discarded wedding veil and cradles it as he weeps her name. This was never in the original script and was ad-libbed by Michael Crawford at some point in the show's original run.
Earlier in the show, when Christine faints at the end of "The Music of the Night", the Phantom kneels beside her and takes her hand as he sings the final lyrics. This too was not part of the original script and was improvised by a Phantom during a production in San Francisco. When word spread about how well this gesture was received by the audience, it quickly became part of the standard Phantom performance.
Originally Christine only kissed the Phantom once and then just hugged him, but during one performance Michael Crawford's makeup got stuck to Sarah Brightman's lips and came away with her! She had to lean in again for another kiss so he could quickly reattach it. Again, the second kiss was added to the blocking and has stayed there ever since.
An inverted example: In the original London production, the Phantom catches Christine as she faints and carries her to the bed (see the Music Video that was made for "The Music of the Night"). This was initially done in the Broadway version, but when an actor injured his back during the stunt, the actors' union forbade it. Ever since then, London is the only venue where the Phantom carries Christine. In every other location, she faints and falls to the floor—from this minor detail alone (foreign languages aside) you can determine if a YouTube clip of this scene is of a London edition or elsewhere.
However, as of late they seem to be bringing it back into the Broadway version; starting with Norm Lewis, the Phantom finally catches Christine again.
When the Phantom takes off his cloak and fedora during the title song, he smooths his hair down. Once again, it came about because of Michael Crawford. During the first rehearsals his wig was less slick than it became later on in production, it got mussed up, and he smoothed it down in character. Gillian Lynne, the original choreographer, noted how sensual the pose was and encouraged Crawford to do it again.
Troubled Production: The show underwent much upheaval during its development and preview days — numerous cast changes, backstage bickering over such changes, props and equipment frequently breaking down, and massive overhauling of nearly all the lyrics. Then, just as the show finally debuted, both of its lead actors took ill (Michael Crawford suffered a hiatal hernia owing to the demanding score, and Steve Barton — cast as Raoul — suffered a fall after he replaced him as the Phantom) and then the understudies were knocked out of commission as well.
There were plans for a live-action film of the musical in 1990, to star Crawford and Brightman in their roles from the musical. The original script was being written before Brightman's divorce from Webber, and the project was left in Development Hell for some time before becoming the 2004 film.
"The Music Of The Night" ends on what many fans feel is an ambiguous note—the lights go out, raising the question of whether or not Christine and the Phantom had sex—or more accurately, given her unconscious/entranced state, he raped her. The original draft of the movie settles the question by actually having the Phantom get into bed with the unconscious Christine and for the scene to fade out as he drew the curtain around them, leaving little doubt as to his intentions. This was changed to as unclear as it is in the show—are her missing stockings a Series Continuity Error or something more?—as it would have been hard to continue presenting him as a romantic hero after such a blatant violation (or it would have been too obvious, even to the most naive viewer).
On the casting of the musical side of things, Steve Harley, an UK singer trying to make a comeback, was actually cast to star as the Phantom and went as far as recording the first single of the titular "Phantom of the Opera" song, but was sacked just before rehearsals began and replaced with Michael Crawford. Needless to say, the comeback never happened.
In the late 1990s, Antonio Banderas was being considered for the role of the Phantom in the film adaptation, perhaps alongside Kate Winslet as Christine.
Back in the late '80s, Michael Jackson thought he should play the Phantom in the movie!
Alan Jay Lerner, one of the most renowned librettists of all time and co-writer of shows like My Fair Lady, was going to write the words for Phantom but died when he had only just started.
Why is the Phantom's mask on the poster different from the one seen in the show? It's because that was the original mask design. However, it was soon realized that this 3/4 mask would not only impede the actor's singing ability, but also make it REALLY hard for the Phantom to emote for the cheap seats, so the mask was cut in half.
On the movie side of things: Universal tried to make sequels to both of their adaptations of Phantom, but neither time did it work out:
In 1929, Universal announced plans for The Return of the Phantom, which would have featured sound and, astoundingly, full color throughout the whole movie! Plot details were never given, although Chaney wouldn't have returned (he had switched studios to MGM). In the end, Universal decided to go with the less expensive option of adding dialogue and new sound scenes to the original movie.
After the 1943 movie became a hit, they announced a sequel for 1944 called The Climax, which would've presumably resolved the minor Sequel Hook at the end of the original and feature the return of the Phantom. The Climax actually did come out in 1944 as promised, but all ties to Phantom were dropped, as Claude Rains was unavailable to return. It did use the same sets, though.
The 1925 film was originally much closer to the book, with scenes including Raoul's visit to Madam Valerius, the Perros graveyard scene, and Christine's kiss redeeming Erik, of which only still images remain. Unfortunately the director didn't like this ending (which was strongly supported by Chaney, who wanted to make the film as close to the book as possible), and neither did test audiences, so it was replaced with the mob scene.