Firstly don’t label your daughter as ‘horrible’ she can behave horribly, but as a person she is not horrible. It’s important not to label her like that because it demonises her. In fact she is a teenager, with raging hormones, a scrambled brain and a slightly complex family set up. You were probably also an arsehole at that age, I know I was.
You sound angry with her (and her mother). This is not actually normal - frustrated yes - but your anger at what is fairly bog standard teen behaviour is excessive, and indicates you are personalising things. I would really advise you to pick up a couple of books on teen development and techniques for managing them. Their brains do not work the way adults’ brains do - and you need to adapt your expectations.
In terms of Christmas Day - ask her what she wants and let her do it. It’s not unusual for teen girls to go through funny patches with step parents. She does need to learn to be polite, but if she’d rather spend C’mas day with her mum, let her. everyone will have a better time.
It might be too close to Christmas to do this now, but as soon as you can, when things are calm-ish, take her out for lunch. Explain you can see she’s unhappy right now, and you can remember going through patches like this as a teen. You love her more than anything, but you also want to make sure your girlfriend isn’t being unfairly treated. What would your DD like to be different so you can all get on better? It may simply be less time with your girlfriend, and a bit more time with her mum - so facilitate that. Your daughter is no longer a little girl and may simply not want to have to spend so much time hanging out with a women who doesn’t mean much to her (and a 6 year old).
There’s a theory that part of teenage moodiness is a necessary process of separating from the tribe. They have to stop being little kids to take off on their own, so they can come back later as adults.
Do insist though on good manners when your girlfriend is around, on the basis of treat other people as she’d like to be treated. If she really doesn’t play ball, do develop a system of privileges being removed. But make sure she knows what will happen rather than springing it on her. Do NOT take her presents away whatever you do.
Do really do some reading up on teens. Step back and think a bit about your feelings about her mum and check you aren’t transferring them to your daughter - Dads can have some adapting to do when their daughters are becoming women. Check also that you aren’t expecting her to have an unrealistic interest in your girlfriend and her kid, given she is no longer a little girl.
Above all - remember she will be really delightful by the time she’s in her early 20s. She’s pushing boundaries right now. Hold them, but be there for her - she won’t forget it.