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.
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.
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 into a handsome prince, implying that she just needed to be loved.
Like dementors, the mirrors shards are depression incarnate
Awesome Music: "The Two Of Us Will Always Be One" from the Hallmark adaptation.
Designated Villain: The Snow Queen herself in the original story, as Kai chooses to go with her to her kingdom rather than being kidnapped, and his problem is caused by the mirror shard put within him before he even met the Snow Queen, rather than being under the Queen's control.
Les Yay: Gerda and the Robber Girl; some translations even show the girl later doubting Kai is worth "going to the ends of the Earth for."
The Woobie: Gerda — just look at what the girl goes through!
Iron Woobie: She takes it all in stride and wins in the end.
Another good example of a Woobie in the Snow Queen is Dimly the reindeer from the animated version.
Kai; the kid not only gets a mirror shard in his heart and eye, but the one in his heart begins to freeze his body!