I am guessing that she's learned that at least some of the time she gets what she wants with this behaviour. I also think you are angry and argue with her. When she argues she gets your attention which may be a part of the reason she does this. Negative attention is better than no attention.
Of course you're angry. Of course you sometimes give in to stop the battle. Both are usual reactions. You can't change her but you can change the way you react to her.
I suggest you say no and leave the room or hang up the phone so you won't get sucked in to her fight. She's heard your lectures. No need to repeat them. Learn to be matter of fact, yes or no. Do not try to prove your point to her. She doesn't listen.
You are the parent. It is your responsibilty to be firm and not continue a fight. When you argue with her, you are trying to convince her you're right. You know she doesn't agree. Give up on that tactic. It doesn't work and upsets you.
She manipulates you with her anger. She wants a fight. She has succeeded in making you angry. She hope's that you'll give in to what she wants to end the fight.
All along, you have the power to stop the fight. Say no! Walk away. If you need to leave the house or lock yourself in a room so you won't fight, do it. Separate yourself from her. Say to yourself "I can do this."
This is a new way of handling the situation. You will have to consistently do this, maybe for weeks, because her asking for stuff and you blowing up has gone on for years. Habits are hard to break.
I like the song I'm a woman, hear me roar! This means that you're confident that you are right. You are right to not give in and to take care of yourself. You are the parent! You are right to enforce your boundary. No means no. Stop arguing. Walk away.
I agree that counseling will help. It will be hard for you to not argue. Counseling can help you learn ways to manage your anger. I also agree that you and your daughter going can help both of you how to talk with each other. Your daughter should be in counseling too but I expect she won't go.
If I knew your daughter's age, and history of losses, I might be able to give a better answer.
Re: lying. My daughter lied as a teen. She knew I knew because I knew her and her friends. Same with my grandchild who has lived with me since they were 15. I didn't punish, accuse or yell. I just gave this look that said I doubt that is true. I rewarded the truth.
Give permission based on what you think is true. She says she's going to a friends house. She's lied about that before. My answer was no. Sometimes, I took her and picked her up.
I made it comfortable for friends to hang out at my house. I got to know them. I played attention to what the did and how they acted. I never said someone couldn't come over except when my granddaughter asked me too.
To parent teens one has to respect them even when the disrespect you. I refuse to argue. I walk away. My action shows them I won't be disrespected. No need to lecture. She knows she's been disrespectful. I'm the adult responsible for teaching her how to behave. I have power. She doesn't.
Along with using my power, I consciously give her power in appropriate ways.
Took me years to learn this and hours in counseling, reading parenting books and taking a few parenting classes along the way. Still I fall into the arguing trap every once in awhile. I sometimes make things worse. Parenting is the hardest job we have. To have some success requires a willingness to learn.