I suggest you get yourself into some good counseling to help you do this co parenting creatively. It seems to be causing you a lot of angst and misery. I suspect this was your ex's forte when you were with her, and she has not stopped using these techniques with you because they work. If I make this uncomfortable enough for him, he'll give in. And conflict obviously is something she is capable of and you don't handle well, so she's playing the card to the hilt. And she's 'getting to' you with every hateful communication.
So what you need to do is resolve that you are going to be the best parent you can be, that you are going to be the emotionally stable adult in this situation. After all, your poor daughter is going to have to spend a good bit of her time with this other person and you don't want her to be left at the mercy of someone who loves conflict and drama (your daughter may be like you, you know, and sensitive, and may have a hard time with it too.) You owe it to her, since you did have a child with this mean drama queen, to be there for your daughter, to claim your every legal right and provide her with emotional support and help her deal with her mother creatively. To do this, you need to work with someone who can help you deal with her first.
Follow the court order to the letter. Be careful to save all evidences of her efforts to torment you and be ready to take her back to court at the first real sign of refusal to follow the orders. You know she will push and push and try to "up the ante" and that she will SAY anything she possibly can to get to you and upset you. Until you stop reinforcing her, she won't suddenly become more responsible or bearable. So see someone to help you work through this as best you possibly can, and be constantly vigilant against her pushing. Do not give in to her, do not let her mean conversation and her tendency to make a huge deal out of every single thing get you to quit trying or keep you continuously upset until you give up your efforts to parent your daughter. Get counseling.
As soon as she's old enough, have your daughter in counseling when she is with you on your time. That will be more valuable to her than any amount of extra curricular sports, etc., stuff that mom is pushing on her time.