These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Though he is a thoroughly unpleasant man- and an attempted rapist to boot- it can be quite hard not to feel sorry for the Duke. Satine did, indeed, make him think that she loved him, and she treated him spectacularly badly thanks to her infatuation with Christian. Doesn't excuse the rape and threatened murder, but the Duke could be considered more sympathetic if you see things from his side.
Essentially, Satine is a prostitute who manipulates her client into spending a ton of money on her and on a musical that covertly mocks him. She sleeps with his employee instead of him, and everyone covers up for them and mocks him behind his back.
Was Nini Legs-in-the-air a complete Jerkass for giving away Christine and Satine, or was she entirely justified? Considering everyone was relying on Satine becoming the Duke's consort so he would fund the Moulin Rouge, and the couple just seemed to rely on everyone to keep the secret for them without being exactly discreet about it themselves, it's amazing the Duke didn't find them out in the first place.
It's heavily implied she's motivated by jealousy, as everyone covets the "Sparkling Diamond" Satine, whereas Nini is just another Diamond Dog.
Is Christian finally experiencing One True Love and his heartbroken behavior at the end expected, or is just Puppy Love and he loves the romantic idea of the them, and thus his humiliation and degradation of Satine when his idealistic worldview of a prostitute is shattered unforgivable, and proof that it was?
Awesome Music: "El Tango De Roxane", "Elephant Love Medley", and "Come What May".
Covered Up: Many don't know that "Lady Marmalade" was originally done by Labelle in The Seventies. Also, in spite of its popularity and the fact that it was used in a good deal of the film's marketing, that was not the version actually used in the film. The song itself, blended with Nirvana's "Smells Like Teen Spirit" and original verse spoken by Jim Broadbent, is the first really big number in the film.
Esoteric Happy Ending: It's really, really hard to find the good in the situation at the film's opening: Satine dead, Christian almost broken with grief, the Moulin Rouge a closed-down, decrepit ruin. However, the absolutely epic triumph over the Duke infers that the protagonists' ideals go on, even if they themselves are dead or scattered.
Hilarious in Hindsight: The songs "Diamonds Are a Girl's Best Friend" and "Diamond Dogs" are both featured in the film's soundtrack. Fast-forward to the My Little Pony: Friendship Is Magic episode "A Dog and Pony Show" in which both songs are referenced in close proximity to each other.
Ho Yay: Defied or lampshaded, when the Narcoleptic Argentinean admires Christian's attitude. "I like this boy..." ( Beat, the Bohemians give a weird look) "Nothing funny. I just like talent."
He also accidentally (?) touched Christian's crotch as he said it.
Love It or Hate It: There really seems to be no middle ground with audiences. Either it's a festive, magical experience with beautiful set pieces or it's schmaltzy dreck that does a complete disservice to modern music.
Narm: Some people might find Christian's hysterical sobbing/laughing when Satine dies a bit...misplaced.
Word of God is that this is deliberate; Christian is so utterly devastated by Satine's loss that he's past caring whether or not he sounds ridiculous. Whether or not the audience agrees is up for debate.
The movie's constant preaching about love is this too, especially combined with most of the performances in the movie.
True Art Is Incomprehensible: At first played straight, then inverted. By Word of God, the reason the Bohemians couldn't get their version of "The Sound of Music" right and resorted to Sesquipedalian Loquaciousness, and why Christian could get it right and wow them, was because Christian had an understanding of True Art that they did not—it was a stylistic choice. This is then continually inverted throughout the whole movie, where modern songs are used in place of the sort of music there would actually have been in 1899 Montmartre in order to help the viewer understand what it would have been like then and how those going to the Moulin Rouge would have felt (their idea of decadence and ours would not mix, but that is what it was to them), as well as to express feelings the characters otherwise could not. It's an interesting conceit, but whether it actually works or is appropriate is up to the viewer.
Once Acceptable Targets: The dance scene at the very beginning opens with a tiff between La Goulue and Aicha, an African woman whom the former refers to as "that dirty-necked Algerian"; Henri seems to take it as a joke, and somewhat defends Aicha.