What is the ruling on a female doctor delaying the treatment of a patient, leading to his death? And does she break her fast during her menstrual period when fasting for two consecutive months? And if she breaks her fast one day out of forgetfulness, does she complete the fast or start anew?

If the person mentioned passed away due to their illness, and their death was not caused by your action, then you are not required to offer any expiation (kaffarah) or blood money (diyah). The obligatory fasting that requires continuity is not interrupted by menstruation. If you ate or drank forgetfully while fasting, then there is nothing upon you, and you complete your fast.