Egypt is full of hospitals, but it cannot be said that they are all reliable. Your embassy should be able to help you out in case of an emergency. The most common health problems in Egypt are often related to the heat—ranging from prickly heat to severe sunstroke. Heat related illnesses can be prevented by using sunscreen, and other sun-protection aids, drinking water at regular intervals (without waiting to feel thirsty) and in the case of severe dehydration, administering an Oral Rehydration Solution (ORS) to the patient. The other common health risk is bilharzia or schistosomiasis—this can be prevented by avoiding swimming in fresh water bodies. Rabies is quite common in Egypt, so do take the requisite precautions, and in case of an animal bite, seek medical help immediately. The ambulance in Cairo can be called by dialing 123 or 77 0123. However, you should be prepared to wait.

Generally speaking, Egypt is safe for travellers. However, a few precautions will not hurt—women are advised to observe modesty in dress and behaviour. There are innumerable stories regarding the annoyances women travellers face—these range from obscene remarks made in public places to pinched bottoms and, worse, bodily contact in crowded places. Rape and serious physical assault are rare. If harassed, try to completely ignore the offender—this usually works. Travelling with a male companion who can be introduced as your husband is also effective. Also avoid deserted places and walking alone at night. If things get serious, do not hesitate to contact the police/your embassy.
As far as crime goes, pick pocketing is a much more likely scenario than serious crime.