I think that calling a woman "a sleek bit of mink" is damning of her personality and motives. If the person who wrote this knows anything about mink, he/she is saying three things: 1. The woman is pretty 2. She's a bitch; and 3. Stay clear of her. Note that he/she said "She's a sleek bit of mink", not "She's as sleek as a mink". There's a very big difference. In essence, he/she is saying that she is a mink. What does that mean? Mink are highly territorial, highly aggressive animals that, among other things, will attack and eat other mink. I have wild mink on the salmon river where I spend my summers. On the surface, they are cute, and in the water graceful. In reality, they are fast-moving, highly opportunistic predators. I think that the person who wrote the sentence you are asking about knows this (it's pretty common knowledge), and that his description of this woman is an indictment.

I don't think this is a real idiom. I believe it's been made up for the purposes of the Game of Thrones universe. Apparently it is from a scene where the women are getting ready as the Queen is coming and the expression is about the Queen. As it has been made up we cannot give the real world meaning, as none exists, but only guess at a meaning. If you want to know what it means, then perhaps observe the Queen and how she reacts to the other people. Possibly it could mean that she has some characteristic in common with a mink... Being wild, or being attractive, perhaps attractive and vicious? Alternatively, I can tell you that the French subtitle for this scene is apparently translated as 'coquette' which means 'flirt'. Perhaps you could look at how they translated that scene for Mandarin to see how they translated it? Does she flirt, or take many lovers?

It probably means 'she is very pretty'. A mink is a small, furry animal like a ferret. They are famous for being used to make fur clothing like scarves. Often we use the word 'mink' to just mean a piece of clothing made from mink fur. Sleek can mean a few things - in this case it means soft and smooth. A 'sleek mink' is a fur piece of clothing that's very nice to touch and very fashionable. Calling the woman a 'sleek mink' means that she's very pretty. It is probably not a nice thing to say - we are comparing the woman to a piece of fashionable clothing, instead of describing her as a person.