She's acting out. This is very common in divorce, and kids usually level the complaints at the parent with whom they live the most. But she's also having this problem at school, so it's not entirely that she doesn't respect you - she's just having problems adjusting.
I think you have to separate out the issues and behaviors that are crucial, and those that are not. If she doesn't take a bath every day, is that horrible? If she mispronounces a word, so what? Instead of correcting her, try to use that same word in a sentence later on. Don't force her to make her bed right now, or similar issues, but don't wait on her either. If she needs to do certain things to be sure her backpack is packed or her lunch & snack are done, then stick to those that really matter. If she goes to school without her homework because she didn't listen to you, then she will endure the consequences from her teacher. If she's hungry due to no snack, she will survive. And she won't do those things the next time.
If she is mouthing off to you, walk away. Tell her you will talk to her when she's in a better mood and able to be respectful. Don't respond with anger and yelling because that's what she's doing. You don't want to engage in the same behaviors you object to in her. That's a huge reason I don't believe in spanking - it teaches them that hitting is okay if you are bigger. But you don't have to listen to someone who is disrespecting you - and that's a good lesson for anyone to learn, but especially girls. Girls and women put up with too many put-downs! Don't rely on her father to correct her - if he doesn't respect you, that's probably a big reason why you are divorced.
In addition, her lying and screaming are getting her plenty of attention - so you have to make sure there is no reward for her. Take away what she wants - which is attention. It's only going to be a short time before she wants something else from you - attention, a favor, a meal made, etc. I used to tell my son, "I'm not your maid. I'm your mother." And while it takes time, it's okay for kids to see that the more they pitch in, the sooner they get what they want, which is more fun and more time with Mom. If a child goes to school without her favorite shirt washed, that's okay. And it's even better when you say, "Oh, is that what you were talking about? You were yelling so much that I couldn't hear you ask nicely. Oh well, maybe next time you want that shirt, you will pick it up off your bedroom floor, put it in the hamper, and bring the entire batch of dirty clothes down to the washing machine." Or, "Maybe next time you want a particular snack, you will put it on the grocery list yourself." And don't forget, "I'm sorry, I don't buy X for kids who think it's okay to yell at me."
The important thing now is to de-escalate the situation. It's okay for your ex and you to have different parenting styles. It's not okay to allow disrespectful talk about the other. You can't control him. You CAN control yourself and what you permit in your own home. It's okay for kids to be told that Dad's rules go on in Dad's house, and Mom's rules apply in Mom's house. It's okay for kids to DISCUSS things that bother them, but not to scream and rant. Sometimes young kids need help verbalizing their feelings, so you can help them along with prompts like "I want to understand what's bothering you." (Note that you don't say "I want to fix what's bothering you.") Try to get them to start at the beginning, to take a breath, and to put the problem into 2 sentences. Then repeat it back to them in your own words, and ask for more details in 2 more sentences. There are some good books on how to talk so kids will listen and so on - ask the children's librarian at your local library for some good, and free, suggestions.
When Dad's permissiveness spills over the school situation, it's up to him to go to the teacher conferences or field the calls from the principal. Be sure that he is hearing the problem FROM THE TEACHER and not from you. If the teacher calls you, ask to put your ex on a 3-way call. If you get an email from the teacher, forward it (without comment) to your ex and ask her to cc him in the future.
Not that this is your precise problem, but a friend of mine handled her kids' bad language with the following: they were not allowed to use certain words, but when they pointed out that they heard the parents using them on occasion, my friend said, "You're right. We do occasionally. Dad has 'driving words' and Mom has 'cooking words'. When you can do either of those activities on a regular basis, you may use those words!" Worked like a charm!