leelee, your daughter loves you no matter what. If you are sharp with her, she will still love you. If she perceives you to be the meanie, she will still love you. She is at an age where they say what they're feeling, which is GOOD. My 3.5 yr old granddaughter used to say, 'I don't yike you, grandma", and then look at me for my reaction. I would just say, "Oh well, you'll get over it", and sure enough, she did.
It doesn't really matter what happens at her dad's house. 1) you can't control that, and 2) she'll behave one way there and another way at home. Imagine if she was just living with him and you weren't around. Do you think her playschool and teachers would put up with the nonsense she's giving you? Not a chance.
Your daughter is challenging you. My granddaughter does that all the time to my daughter. My daughter is a social worker, but she's not a child psychologist. You just have to ignore her, stand firm, time outs, whatever... but don't think her psyche is being all that affected by what's going on. She will, however, as Hawk said, learn to play you, and all you have control over is your house.
You should google the Montessori website on challenging children - they have 17 wonderful ways to stop it.