There are some Kindle version formatting issues with this version, I'm afraid. There are a few points at which there should be poetry or quotations (italicized, indented text) but they are missing. Not a huge problem, but this means a long poem is missing from "The Shoes of Fortune" and the punch line of "The Old House" is lacking.
Aside from that, I found this collection uneven, and remembered why as I child I largely preferred Grimm's Tales.
There are a few real winners. I forgot how much I like "The Snow Queen". It's a long meandering tale with plenty of intriguing minor characters and the edges of lots of other stories surrounding the core. The scene in which Gerda hears the story that each flower knows was beautiful and very surreal. I adore the side plot about the vicious, but ultimately kindhearted, little robber girl. The ending is sort of a let down, sadly, but only because the first 3/4 are so good.
I also really enjoyed the other long one: "The Shoes of Fortune." It consists of a series of sub-stories about the eponymous shoes, which cause the wearer to receive whatever they wish for. Unlike in a more traditional fairy tale, the wish-receivers have no knowledge that it is the shoes that are affecting them, and the overall moral is 'be careful what you wish for'. The second section is especially amusing, in which a townsperson who had thoroughly romanticized the middle ages finds himself back there...much to his dismay.
"The Fir Tree" is moralizing, but has quite lovely use of language. "The Emperor's New Clothes" is a classic. "The Shadow" is proto-Twilight Zone. I approve of that.
There are quite a few more that are up and down: either not that interesting, or simply too odd for me. A few are quite creepy, and a few like imitations of older stories.
Then there is the one in which Cupid is represented as a "naughty boy", that one should attempt to avoid, although usually in vain. It was so bizarre that I hopped over to Wikipedia for a bio, and, sure enough, Andersen was plagued by unrequited love. No surprise there.
Then there are a few that I hated. Ones like "The Story of a Mother" or the infamous "Little Match Girl," or even "The Red Shoes." They are well written, often pretty, but I am personally not okay with stories, ostensibly for children, in which the end is about how beautiful it is when poor children die horribly, because they go to heaven. Not okay. That is some pernicious and nasty stuff there.