By analyzing the information you have provided, here are the major possibilities, concerns and the things you should do.
First thing I would strongly recommend you to meet a Breast Surgeon without any delay. The major concern here is as you said a very rare possibility of Breast cancer. Your and the description are not very typical but its always better to be in the safer side.
Changes in the breast size or shape, abnormal swellings, nipple retraction, nipple eczema which is very persistent, bleeding from nipple, lymph node enlargement are all good reasons for you to be referred.
That’s said there is no reason to be alarmed. I just think you need to be referred and got to have a better assessment. A surgeon will be better equipped in this regard. Since you were not satisfied with your GPs clinical examination and treatment its better if you directly get in touch with a Breast surgeon. In case if you need referral in your country, you can ask your GP to provide a referral letter.
Other condition that can cause the above said symptoms is mastitis itself. But since you are not pregnant or breast feeding mastitis is less likely.Sometimes a fungal infection can occur in breast tissue and this could give you the itch, rash and glands under your arm. It would not respond to antibiotics and may need treatment with quite a long course of antifungal tablets.
Please feel free to contact us for any follow up questions.