1) her condition was deteriorating rapidly
2) overlooked = missed. The symptoms were not noticed, or if noticed, no attention was paid to them.
3) tossed out = rejected. For whatever reasons, those papers didn't meet the criteria of the authors' study.
4) 'how they fared' refers to the outcome of their disease and its treatment. In general, if they fared well, they got better. If they fared poorly, they got worse.
5) personalized = individualized. It is recognition of the fact that no one treatment is best for all patients. In this age of genomic medicine, we actually take into account the genetics of the cancer (e.g. BRCA1, BRCA2, HER2 for breast cancer) as well as the genetics of the patient with regard to their expected metabolism of the medications to be used.
6) 'Sex' here means gender. Nobody is treaty cancer with sex. Too many patients would sign up for that study.