The Snow Queen's reason for abducting Kai is never explicitly stated. While she probably just did it so there'd be a plot, it's not uncommon for dramatized productions to make her a lonely Anti-Villain who wanted Kai for a son, which does certainly make sense given her kind treatment of him in the original story.
Likewise the Hallmark adaptation has the Snow Queen as the villain but in the end she gets a happy ending as well when her Polar Bear servant is transformed back into a handsome prince, implying that she just needed to be loved.
There is an animated version where the Snow Queen herself was under a curse, and captured Kai because she needed his help to solve the puzzle in order to break the curse.
In Shelley Duvall's Faerie Tale Theatre, the Snow Queen is protecting Kai and providing him with the opportunity to save himself. In that version, the troll's laboratory is on a raft floating around in outer space, and at the end the Snow Queen is shown to blow her wintry breath and send him sailing off far away.
At least one adaptation made the demon/Satan/troll that created the mirror doing so to show mankind how rotten it is, and had a My God, What Have I Done? moment when it broke. Of course, it almost certainly isn't what Hans Christian Andersen was going for.
Like dementors, the mirror's shards are depression incarnate.
Is the Robber-Girl a budding Psycho Lesbian with an unhealthy interest in Gerda? Or is she a lonely — if maladjusted — child seeking companionship with someone who isn't a criminal? Is she some combination of the two, a young girl with a crush but lacking the knowledge of how to express such feelings without resorting to threats and/or imprisonment?
Awesome Music: Among many versions the BBC's version with the songs makes great use of singer Sydney Rae White.
Common Knowledge: "The Snow Queen" is frequently (at least in the English-speaking world) referred to as Andersen's longest story, but "The Marsh King's Daughter," "A Story from the Dunes" and especially "The Ice Maiden" are all longer.
Gerda goes through a journey worthy of The Hero's Journey, all to find and return with her Playmate, Kai. This is all in spite of him having been mean to her due to the Mirror Shard's influence.
Gerda saving Kai from being permanently frozen. When Kai finally recognizes Gerda after being free of the Mirror Shards, he immediately goes back to being her friend, and Gerda holds no ill will against him.
Also doubles as a literal example.
Iron Woobie: Gerda takes it all in stride and wins in the end.
Les Yay: Gerda and the Robber Girl: when she gets a look at Gerda, she demands to have her and sleep with her. Some translations even show the girl later doubting if Kai is worth "going to the ends of the Earth for."
Toy Ship: There are a good number of fans that pair together Kai and Gerda. This extends to several adaptations, where they end up becoming a couple after the events of the story, and after they become adults.
