ANOTHER TSQ-IV DEPICTION

In this steampunk version of the Andersen fairytale, the Clever Princess is played by Cindy Tim (an Asian character actor as a red-haired Southern belle...), while her fiancé is the Othello-like Rhett George (here, in a soft brocade nightgown).

Another red-haired Clever Princess (though not toppling auburn as most frequent hair colour!) and some positive discrimination... seeing both royals played by these actors was rather unexpected!

Said princess decided she fancied a husband one day and extended the invitation to any eligible bachelors around. Any who could speak well and comfortably to her would be her husband, and the one who succeeded, a rugged chap with long hair.
... prince and princess waving a tearful goodbye. 
On Anna in Frozen: She herself is quite similar to Andersen’s husband-hungry princess, and at a push you can see the rugged Kristoff in the prince.

"A story about a lonely Princess.  Men come from all over to win her hearts, but have no such luck. Suddenly, there was one who went to the princess not to woo her, but to hear wisdom. The Princess and the youth (Karl? Klaus?) were taken with each other, and so he stayed.

"a princess of surpassing cleverness and beauty":

The Princess

The princess is a supplementary character who only appears in one of the chapters of the Snow Queen story - but she is nonetheless an admirable and inspirational female character, whose story hints at a much longer and grander untold narrative.

According to the Raven:

"In this kingdom in which we are now sitting, lives a Princess, who is so immoderately clever; but then she has read all the newspapers that are in the world, read and forgotten them again, so clever is she. Lately she was sitting on her throne, when she began to sing, and the theme of her song was "Why should I not marry?" "Well there is something in that, she said, and so she determined to get married; but she must have a husband who knew how to answer when spoken to, not one who could only stand there and look grand, for that is too stupid."

What a fantastic introduction to a character! And what a refreshing change from the fairytale standard of princesses being first and foremost beautiful! The Princess decides on her own that she wants to get married, and she then goes on to specify what kind of a husband she is looking for - one who is intelligent, unabashed by royalty, unafraid of her power and one who "feels at home' with her.

The Raven then goes on to describe the meeting of the princess and her husband-to-be:

"He was gay (merry) and well behaved, but had not come at all to pay court to the Princess, but only to hear how clever she was. He had every reason to be satisfied with her, and she no less so with him."

Again, what a refreshing subversion of the princess trope! The princess' chosen husband is bright-eyed and merry; a poorly dressed "little person" - a wanderer with creaking boots and a knapsack on his back. No dragonslayer he - but one who can match the princess' intellect, rather than impress her with feats of arms.

As well as being clever, the Princess is later shown to be generous and sympathetic, willing to help Gerda into her new golden carriage herself without formality. Though her part is small in the overall story, and she and the Prince go away to "live in foreign places", one feels sure that such a great character must be the heroine of her own legend.