Personally I like summer for Egypt. Early for the south and mid/late for Sinai and Aquba. Some find it too hot.
For the sudanese Red Sea, I preffer the spring months.
Don't know about Djibouti, Eritrea and Yemen, but I would guess it's different.
Re: suit. Water temperatures vary from 20 to 30 degrees centigrade depending on location and season. If you do one or two short dives a day, most can get away with a shorty in the warmest periods. For liveaboard diving where you spend 3-4 hours a day in the water, I would reccomend a full 3mm or 5mm suit. In the colder months in Egypt, many dive in dry suits.
good luck