I want to commend you for being such a strong mom for your D12. I break down regularly even though I have a strong support network (Slew of free therapy, supportive family, and friends). I think it's amazing that you are able to do so much for her and continue to stay strong while she teams up with the immiture parent to bad mouth you.
(best emoji ever) we need to have a one-time, members-only meeting to meet globally somewhere and just "hug it out"
So I would like to offer some random tidbits that has helped me with my D12. Only you know the situation well enough to decide what can work for you in your area or if any of this is relative to your situation and would even help:
Check with a lawyer (maybe pro bono) and set her up with some sort of counseling. I spoke with the school guidence couselor to email me, recommending my D12 should see a therapist, so he couldn't refute it. She is a tween dealing with her parents divorce and that should be enough to recommend some sort of periodic couseling. This is an unbiased person that can talk her through situations and express her feelings without mom or dad finding out if she doesn't want to share. She needs a place to feel safe to badmouth both parents if need be. Maybe also get her a journal that she can use with 100% complete knowlege that you will not be able to open/find it.
She does need to be a teenager and may need to hate you. She is trying to branch away from you to discover her own indepenance, but clinging to her dad because she's scared of growing up/doesn't like the rules that come with it. It might not be ideal, but at least know that it might be normal for her situation and know that there is a mix of her trying to be a teenager and siding with her dad. She called you a slut because she wanted to hurt you. She is finding ways to break out and test her independence, but chose that word because she knew that word pushes your buttons (or lying about her dad telling her because that would push your buttons). Maybe not this time, but any other time.

Sometimes I tell my daughter that she forgot to slam the door

I also research and talk to friends about all her little behavioral quirks and how to help her through them. Find adult friends with D12s that are acting out in homes with no PDs. That might help you determine relate to her as a tween that's rebelling. She will also be able to bond with a friend that is lashing out in healthy ways and not all the time. Who knows, you might be able to come close enough that they can take your daughter birthday shopping for your with their daughter. I would always try to convince my friend's daughter to buy the tackiest thing for her, so she would have to pretend to like it and wear it.
I found out D12 only likes hugs from the side, she'll cuddle with me only if I act annoyed by it (or I act overly childish about it), loves to pretend to annoy me if I'm overdramatically upset when I respond to her, and she's more open to talk if she's doing something else at the same time (hanging out by the campfire, board games, sitting by the pool). I ask her personally about what she wants to do with me (when I'm not vulnerable to being shut down). She still turns me down a lot, but we get some bonding time in there.
Ask the lawyer if you can install an ipad app to forward messages to your email (you will have to decide if it is worth D12 finding out). Just to keep and show parental alienation in court (not to talk to her about), maybe to suggest parent counseling, supervised visits, or something?
Try to set limits that are more general and don't make them about her dad. Only allow 1hr/day ipad time or something, if she breaks those rules, setup a child's login account with a curfew (only after she breaks the rules). Do not allow any phone calls after a certain hour or to disrupt unless it's an emergency (she can talk until dinner's ready or until the movie starts and she will have to call them back after). Remind her to call back after because that's what people should do. When she says she gets to eat out or stay up late at her dad's house, tell her that there are different rules at different households. If she says she doesn't like your rules, respond that she has a set schedule because consistency is important for children (sometimes I say teenagers because my D12 likes when I group her with teenagers). Don't argue with her when she says "then why does dad... ." only respond that when she is in different households, different rules apply (also, you can reinforce by saying she needs to be extra respectful as a guest in a friend's household). If you are consistant with rules and attribute the rules to her well being, then she is more likely to see the rules as separate thing you have less control over and non-negotiable.
And most importantly,
take care of yourself, be the person you want your daughter to turn into. You need to do things for yourself, have healthy relationships with adult friends, date if you want to date. Show her how she should act/react if she is ever in that exact same situation when she is your age (I hope that would never happen). Show your daughter that you don't succumb to bullies, that you will assertively tell the people you love that by disrespecting you, they are being is hurtful. If she calls you a slut, tell her that "slut" is an inappropriate word for her to say and punish her. If she asks why dad is calling you a slut, tell her that slut is an inappropriate word and that is something you will need to email her dad about (document, document, document). If it ever gets to a point that she tells you she doesn't want you to email her dad because he will punish her for telling you, then tell her you won't email him and are glad that she is sharing with you (that might be the time to talk to a therapist about slowly introducing PD in an age appropriate way).
I really do hope some or any of this is helpful, I always feel like I'm rambling/ranting. I feel your pain because you are not getting any sign that she will understand what her dad is doing. My biggest fear is that my D12 will get sucked in, painted black, and stuck. How can I expect her to notice what's happening when I got sucked-in by him as a grownup and I had a healthy upbringing? I think it's easier for teenagers to catch on when they have both households to compare. Do everything you can to show her what a healthy family is (even if it's just you two). She will get there, and anytime she starts to seem confused by what's happening with her dad, know that she is getting there and replay that small "win" in your mind over and over to give you strength.
My D12 asked me why I'm trying to keep her away from her dad. Her dad sent me a hateful email when I moved to the next city, telling me I did this to keep her away from him (I did not move far enough to change weekend visits with him). The intention of this email was to demand more time with her in the summer because he was moving 3 states away (a fifteen hour drive one-way). Anyone know how this is my fault?
I told my daughter that I was glad she asked me and it's really important to communicate, especially when you are confused about something like this. I went over every move that I made, every move he made, and the reasons for it. I could tell in her face that there were discrepancies with his story and this is probably my only real "win" for me so far with her. I think about that as often as I can because the look she had on her face means so much more to me than all the therapists and family members telling me "she will figure him out eventually".
Also a heads up with the child support, any backpay is your money, your ex is paying back a debt he incurred to you when he didn't pay past child support. Any current child support is to support the upbringing of your child, not what the child wants. Finally, talking about child support with your child is inappropriate. I had to listen to my D12 tell me that I should buy her a playstation with the money her dad gave me for her (it was backpay and yes, she said the amount he gave me). I was not ready and lost my cool, I'm pretty sure I should have said "I am sorry your dad told you how much money was used for child support, it is inappropriate for him to be talking to you about this, and this money is for your care that is predetermined for us. We use the court system when we can't agree about what is best for you." I think I will save the response for a new thread... .
P.S. "0" Days since last incident... .

... .and it was going so well for months... .