OP, and I mean this kindly. You are going too deep in to this. This is obviously about so much more than than her not texting you on Christmas Day to say thank you.
Its now evolving in to 'and she lied saying she had tried to call and was sarcastic'. The drama is still continuing.
If you would have just left it on xmas day this wouldn't be happening. So you do need to look at your own behaviour in this.
Teenagers can be incredibly rude, selfish, ungrateful and all that jazz - because they are not adults yet and they only think about themselves. However you was brought up doesn't mean that she has to follow on the same path. You are two different people.
This relationship is still relatively new, one that you are clearly putting a lot of effort in to but now you feel rejected or taken the piss out of because a teenager is not returning the same amount of effort. Deep feelings over your dad are also going to be tangled up in this.
My eldest is 25, over the years she has shown some not great behaviour - no one is perfect. I also worked with a teenager who was literally living on another planet. I used to pick her up on the way to work and drop her off as her house was on the way, when I took a weeks holiday from work she text me in the morning asking what time I was picking her up - she actually thought I would still be taking her in on my days off. She would lie about making mistakes even though it was clear she had actually done them, then she would go bright red and cry in the stock cupboard and phone her mum.
Honestly some teenagers are from another planet. Who they are now is not who they will be when they are 25.
But - you need to look at why you still bought this girl a coat that was too expensive and why you felt you couldn't say no. And then why you got upset on Christmas Day because she wasn't quick enough with a thank you. This is really all about your feelings - not her behaviour.
Also its totally ok to walk away from all this but dont let it be over a coat - because its not really is it?