I've only read your first sentence and have 2 comments. It's her job as a 13 yo to hate you. lol And, you are making a big mistake going out of your way to make her happy. You are her mother and your job is to ready her for life as an adult. Her happiness is her responsibility. Your responsibility is to give her a normal life while teaching her that life has requirements and consequences when she doesn't meet those requirements.
Yes, you are concerned for her happiness and it is your responsibility to make it possible for her to be happy but it is definitely detrimental to you, her and your relationship to go out of your way to make her happy. This gives her a sense of entitlement which then does cause her to be unhappy when she doesn't get everything she wants whether is is physical goods or always her own way in doing things.
I suggest that she is a good kid because others do have boundaries with her. She knows where she stands with them. She knows what they expect and knows the consequences of her misbehavior.
I've struggled with boundaries my whole life. It took me at least a year as an adult to even understand what boundaries were. Your daughter is acting out because she needs boundaries. She needs to know that you won't go out of your way to make her happy. She needs to know where she begins and you end. I suggest that you find some counseling to help you develop boundaries for yourself and with her.
In the meantime here are some basic issues. Do not buy her everything she asks for. You decide what is appropriate for her to have and what will fit into your budget. Develop a plan so that she works for some of the things that she wants. This is where some rules will come in handy.
You decide what sort of behavior you are comfortable with. You write down a list of chores that you expect her to complete. Then you talk with her about these and be willing to make some compromises. Let her choose a set number of chores to be completed by a certain time. Depending on how difficult the chores are 3-5 is a good number at this age. Examples are everything from making her bed every morning to doing the dishes, vacuuming, etc.
Also you decide on what values you want her to learn. Values are such things as honesty, respect. Respect is an important one because it includes such things as speaking in a respectful tone of voice. Doing what one's parents asks them to do without talking back. Many of the behaviors with which you're dealing is related to respect.
First you have to respect yourself. Do you put up with her behavior because you suspect you're not worth as much as she is and therefore deserve to be talked back to? This, appears simple on the surface but it is not. Took me years, and I'm still working on it, to realize that I'm a really good person and deserve to always have respect.
Going along with that is the idea that because I deserve the respect I do not need to get angry in order to get respect. Respect is a fact of my life. If my child is not respectful they must leave my presence until they can be. I tell them this in a calm, matter of fact tone of voice. "Go to your room until you can speak respectfully to me."
I think that one reason I used to yell alot is that unconsciously I felt that I had to convince them that they needed to respect me. I don't have to convince them because there is a consequence to them for being disrespectful. The consequence will either convince them or not. If it doesn't they keep getting the consequence. They have a choice. Respect or not. No matter what they do or don't do, I deserve respect. This forms a boundary between the two of us.
Often the fights we have with our kids is about that boundary. We are defending the boundary and taking on the responsibility for the child to acknowledge it. We don't need to defend it. The boundary is there and it's up to the child to learn that the boundary is there and you will not change it.
I suggest that your daughter is constantly testing to find out where your boundaries are. I suspect that you frequently do not know yourself where they are because you are focusing on making your daughter happy instead of helping her to be an independent person, separate from you, who knows what is expected of her and what the consequences will be if she doesn't come close to meeting those expectations.
I empathize with you. I went thru this with my daughter. It's a miserable place in which to be. And it does take some serious work on the part of the parent to dig out of the hole. The work is painful but worth it.