The core of the idea, for me, is that it is a cold-centered subclass that isn't screwed or just engaging in a pure arms race* when facing a cold-immune creature. To do that, I've had to make one sweeping assumption and assertion that... definitely might fall apart on thorough investigation. Specifically, I'm positing that a sorcerer tied to Winter could have some kinship or authority in relation to all naturally cold-resistant or cold-immune creatures, as a stand-in for the fact that there's no Cold Subtype in 5e. If this doesn't work for your setting, you could rework relevant abilities to suit your cosmology, or possibly the Winter Kin just isn't a great fit for your game.
*Arms race: a cycle of one-upsmanship in which this character's cold is so intense it can ignore cold immunity, which opens the door to that monster having Super Cold Immunity that still applies. It's a more valid approach than my comments here suggest, though. It has a lot of benefits for pure clarity in playstyle.
Winter Kin Sorcerous Origin
Your sorcerous power comes from a mysterious time in your life, when you disappeared into a blizzard, the ice broke on a frozen lake, or you went alone into the snow and your tracks just... stopped. The next thing you know, some time - anywhere from a day to a few years - has passed, you haven't aged the same amount of time (if at all), and you have a strange command over the frost and creatures of winter. Or maybe you just have a bloodline connection to the Unseelie Court or a divine power of Winter.
Cloak of Winter
At 1st level, you gain resistance to cold. As a reaction after taking cold damage from any source, you gain immunity to cold until the beginning of your next turn. Armor of ice and snow briefly takes shape over your clothing and any armor you wear; this does not restrict you in any way.
You ignore the effects of environmental cold, frigid water, slippery ice, and thin ice. Terrain covered in ice or snow is never difficult terrain for you.
At 1st level, when you deal cold damage to a creature, you may use a bonus action to blast the target with gale-force winds as well. Make a Shove attack against the creature, using your spell attack bonus in place of a Strength (Athletics) check.
When you would deal cold damage to a creature that naturally (not as part of a spell or other temporary effect) has resistance or immunity to cold damage, the creature must pass a Wisdom saving throw against your spell DC or become either charmed or frightened, at your discretion, until the beginning of your next turn. On a success, there is no further effect.
Starting at 6th level, when you spend any sorcery points as part of casting a spell that deals damage, you may change the spell's damage type to cold.
Further, when your current sorcery points are less than half of your maximum sorcery points, you may choose to have vulnerability to fire damage. If you do, all ranged attacks against you have disadvantage, and any creature that deals damage to you with a melee attack also takes one point of cold damage for each level of sorcerer you possess. This effect continues until your current sorcery points are equal to or greater than half your maximum, or you choose to end it, whichever comes first.
Starting at 14th level, add your Charisma bonus to the damage of any spell you cast that deals cold damage. If you cast a spell that makes multiple separate attacks, only add your Charisma bonus to the damage of a single attack.
Master of the Frost
Starting at 18th level, you assert your absolute mastery over winter. When you deal cold damage to a creature, you may spend a bonus action to force it to make a Constitution saving throw or be restrained. If it fails this saving throw by 5 or more, it is instead paralyzed. If it fails this saving throw by 10 or more, it is instead petrified (turned entirely into ice).
When you use your Winter Birthright feature to impose the charmed or frightened condition upon a creature resistant or immune to cold, the duration extends to 1 minute. If its Challenge Rating is less than your sorcerer level, you may instead cast dominate monster on the creature without expending a spell slot.
First off: yes, I have created a subclass to support one of the Disney characters most intensely adored by girls under the age of... let's say ten. This does not bother me in the slightest, and if it helps some D&D-playing parent get a child interested in this hobby, then I will call that a solid win. (It's also a fair rendition of a particular Adventure Time character. Also fine, for the same reason.) I wasn't thinking of either of those characters when I first came up with this class, though.
While I'm on this topic: Belle could totally be a 3.5e archivist, Mulan is whatever weapon-wielding class it pleases you to stat her as, Merida is basically already living in a D&D setting, and so on. A team full of Disney princesses could be pretty cool. Were I running that campaign, I would have them get drawn together in a haunted Disney World.With Cloak of Winter, I was trying to implement a superior cold resistance without just going nuts with it. There are probably some available exploits here, such as having friends do small amounts of cold damage to you, but I think they'll prove to balance out in the action economy. Also, this costs your reaction - something that sorcerers don't care about all that much, but that might dissuade a little of the class-dipping to pick this up (for example, from highly reaction-dependent fighters).
For Winter Birthright, I liked the idea that creatures deeply tied to cold, as an aspect of their natures, might be connected in a mystical way that the Winter Kin sorcerer can exploit. Also, that the Winter Kin's spells were more than just cold, but also carried a little of a blizzard's winds.
The first part of Icy Transformation is a way to make sure it's not miserably difficult to deal cold damage at all spell levels, but also not strictly free. The second part is a tactical tradeoff and risk, and that makes it weird. I really liked the early-D&DN-playtest sorcerer that shifted to a different combat style as it burned through sorcery points, and I wanted to draw on some of that idea here without just incentivizing the player to convert sorcery points to slots at the start of the day and have something this potent as an always-on feature. So there's a drawback that makes you choose when to use it, when the situation changes, you cash in a spell slot to boost your spell points to 50% or better again.
Biting Cold is simple, maybe lazy, but getting a damage add to your cantrips at some point just needs to happen. I think there's a reasonable argument that Biting Cold and the first part of Icy Transformation should be bundled at 6th, and the second part of Icy Transformation should come in at 14th. The reason I didn't do that is, well, because I didn't.
Master of the Frost is a pure upgrade to Winter Birthright. The first part is intended to cover the frozen-solid effects that you see in movies and video games, but D&D doesn't usually offer. Outright petrification won't come up often, but when it does you'll feel like a rockstar. Even just applying restrained for the low-low cost of a bonus action is pretty great. The second part felt like the natural upgrade of its Winter Birthright foundation - longer duration, and if you're stronger than the creature in question, also a dominate effect. The fact that you're potentially getting multiple free dominate monster effects in a day is bananas, but there aren't that many adventures that involve fighting a progression of cold-resistant or cold-immune creatures, and when it does happen you should get to revel in it.
There are a couple of highly thematic ideas that got left on the cutting-room floor, never fully developed. One is "I want to make awesome ice sculptures," because Frozen, and the other is "I want to make paths and ramps and complex-shaped walls out of ice," because Bobby Drake. Creation is the obvious spell to use as a foundation for awesome ice sculptures, but it makes real things out of shadow-stuff, and all of the approaches to modifying it through a subclass feature that I came up with were clunky as hell, while not really delivering an effect you'd want to use often. (On a tangential note that crossed my mind while working on this, the presence of water elementals and absence of ice elementals from the Monster Manual is a cryin' shame.)
Likewise, wall of ice is the obvious go-to for Iceman's oeuvre - it creates simple ice architecture, boom, perfe... it's 6th level. Hrrm. Well, I could make the Winter Kin's 14th-level feature some upgrade or variant of the spell. But, holy geez, how am I going to write "do some cool Iceman shit" into rules without it becoming hilariously abusive? Answer: "(expletive deleted)"
Probably the better solution to those design goals is "also you can do strictly cosmetic ice stuff" as a cantrip, about on par with druidcraft, prestidigitation, thaumaturgy, and all of the Elemental Evil Player's Companion spells that do similar things with a single element.