Despite the aforementioned questions, I can offer you some advice regardless of your answers. First, she is a "new" teenager who is acting very appropriately, i.e. wanting to make her own decisions that affect her life, resenting being the one who perhaps has been placed all too often in the caretaker role of her brothers, seeing you as controlling all social aspects of her life,etc..
School IS and should be a highly social outlet for kids. An important life long learning that comes from being in school is how to be a responsible, socially agreeable, cooperative member of a community outside your family of origin. This is an extraordinarily needy time for her socially; there is much she is missing out on. It appears that home-schooling was chosen for her as a "punishment" for her being "too social". I believe she was never given a chance to prove that, over time, she could adjust to the social life of school and balance that with her "job", being a responsible academic student.
I have nothing against home-schooling. I believe, however, that in your daughter's case, home-schooling was chosen as your way of keeping her away from social "temptations". If you can surrender up your fear-based reasoning for a moment, could you entertain the notion of seeing a family counselor with her. The family counselor could hear both of your concerns and then offer up a blueprint for how all those concerns could be fairly addressed. I think you need to show your daughter that you will respect her needs and opinions and that you are open to any plan that would both satisfy her needs and address your worries. I know I'm asking you for a big shift in thinking here but I think it's a path that can allow her to grow up with a stronger, more responsible sense of herself.
