Anxiety is rooted in our thoughts and beliefs. Discover what your daughter's thoughts are and you will be better able to support her. She sounds like a very sensitive child and from what you described, she is already telling you what is going on. "Her heart hurts" suggests that she is feeling pain for someone she cares about (could be herself or someone else).
There is a clear connecction between our thoughts and feelings and our physical symptoms. If you listen carefully to the words your daughter uses you will find the keys to her physical symptoms. Her stomach issues could be related to a feeling of powerlessness. She may be having a hard time breathing because she feels in some way suffocated by circumstances in her life.
Instead of asking her a question such as "What are you feeling?", try using a more direct statement such as "Tell me about the pain in your stomach." If she tries the "I don't know" approach, then just gently say, "Pretend that you know and tell me...". Using the word pretend allows her to connect with a different part of her brain that is less sensored by her inner critic.
It is clear that you already have some very clear, strong answers to her feelings. There are "tough" kids in school that are often being disciplined, she has high expectations set for her schoolwork, etc.
To support her in feeling more empowered you can first listen without interrupting or censoring (don't respond with things like, "oh, I'm sure it's not that bad", or "Well, it is just something you are going to have to deal with" etc.) Then, ask her to give you some ideas about what she could do about the issues. Again, use pretend if she says she doesn't know. Allow her to give you several (I always asked my kids to give me at least three) solutions before you give any suggestions. Also, encourage her to think of any solutions no matter how unrealistic or irrational they might seem at first. You are supporting her in moving from "victim-mode: 'I have no choices'" to brain-storming and problem-solving. This is a process and at first you just want to make the shift not necessarily come up with the "perfect" answer.
Trust your daughter to have the ability to solve this problem for herself. Support her rather than try and fix her. Give her information about bullying and feeling her feelings and communicating, etc. Be gentle with her and trust your instincts. Be careful of getting stuck in your own fear-based thought process and stay curious. Curiosity is the best approach to healing any situation. Curiosity keeps you from telling yourself fear-based stories in your head. Curiosity supports you in gathering tons of information before jumping to any kind of choices.