As a stepmom of 2 girls, 18 and 20, I'm sitting here thinking about how your ex has his hands full. We thought that if the girls got through their teens, high school, they'd pull it together and be kinder, nicer kids to me but it only seems to get worse. They were equally half time in each house until the age of 18. The oldest moved in with us full time and thought she was queen of the roost. When I pushed back (asking for household participation, rules, etc.) She moved to her moms. For 2 months. I was asked to move out, and she moved back in. Her father now has HIS hands full, and regrets asking me to leave, but I have to say, it's good to have your space to take care of yourself.
We'd always heard how great it was at mom's house, and how there were no rules there (Disneyland best friend mom), but comments from mom tell us that's not the whole story, and that they fought with her all the time and played how great it was at our house. I'm a child of divorce too.. it's a sucky situation for a teenager, it just is. Still, don't assume it's any different at dad's house than yours.
Unfortunately young women have to separate emotionally from their moms to define themselves as women. Doesn't mean she doesn't love you, or you her. Trust that the first 16 years of you raising her will imprint on the woman she will become. I was terribly frustrated with the choices divorced kids get that children of intact families don't have, and aren't ready for. Don't bend over backwards. Stand your ground. She's pushing to test her boundaries and you have to keep showing her them.
When our oldest moved back in (she said because there was no structure at mom's house), her dad set very distinct rules about how the house would run. This summer when their mom moved out of town, the youngest moved in full time and started acting out terribly. He finally grew a pair (so to speak) and put his foot down telling her if she didn't live by his house rules she could move to her moms--no car, no computer, no college money, no cell phone. That if she was old enough to not need a parent, she didn't need all the things he provides for her. She started communicating with him and meeting curfews.
As hard as it is for you to let go right now, it is so important for girls especially to have a relationship with their dad. I wonder how she will react when he sets boundaries, because at some point, even if he is very flexible, they will be at odds. Also wonder how she will react to his dating -- or is he married and are there other kids in the home? I see where people say a 16 year old can make residence decisions, and the courts generally support this. It sucks to feel you have no rights as their parents though...and yet are still responsible for them.
You are right though that it's a bad bad pattern to run away from conflict and not learn resolution. Don't give up on that. Not sure what the argument was about, but if it's something you can't compromise on, agree to disagree, but remind her that although she may see things differently, you have the experience and wisdom and responsibility to make decisions based on her safety and well being. No one has asked-- is her dad's house a safe place for her? if not, of course you have to press your custody rights. It's got to be very frustrating to have taken all the responsibility and have your rights not regarded by your ex.
They let 16 year olds drop out of high school too. Stupid but they let them. Give her this chance to bond with her dad. Spend time together.. on your terms. Perhaps set a date for the same time each week. It may mean there are times she has to give up some social event to meet your date, but I wouldn't reschedule at the last minute if she blows you off. You've been so important to her, she's going to miss you, you are the only MOM she has. (it's a special bond even when you're going through conflicts... she probably wouldn't treat anyone else the way she treated you, good or bad). Call her and ask about her day, and just listen. She doesn't get to stop being your daughter : )
You asked how to move on and live this way? Do it well to set an example of how a strong independent woman lives. You love her, you don't need her. You want her to love you but not need you. Be there when your ex calls to find out how to handle her, support him (for her best interest not for his) then let him have to do it! Take some time to enjoy your friends, pay off bills or save for and go on a vacation (you deserve it, and she'll hate being left out, but she made her own bed). Enjoy the peace of your quiet home. Sometimes some space will allow her to bring her best self to your door... you may get to counsel and love her without being "the law".
I hope you can find some positive through this uncomfortable time. It's NOT fair-- as much as you want it to be. Give it some time, and both of them will agree with you that perhaps that night wasn't handled well. (consider maybe he lives with some guilt about his lack of involvement & did exactly what most parents do when their children call for "help" --come and "save" them) I hope he can be more supportive of your efforts as a parent in the future. I think you still have the capacity to resolve the conflict and teach some of the lessons she needs to learn, perhaps with this living situation you can find different ways to be heard.
Don't feel punished. If you are convinced she is loved and safe, think of this as growing pains... it's a dry run for 2 years from now when you really want her to want to be and have the skills to be independent. I'm sure she loves you and someday will be mature enough to thank you for all you've done for her.
Hang in there!
ps if my math is right you were very young when you had her, and saying your daughter was your life meant that you took your responsibility seriously, not that you smother her.