Does the time difference between Cairo and Chicago change during the year?

Yes. Because of daylight saving time the gap moves between +7h and +8h.

When is the best time to call Chicago from Cairo?

There is no overlap between standard working days, so one side has to call early or late.

Cairo is 8h ahead of Chicago. This gap is not fixed: daylight saving time moves it between +7h and +8h during the year.

Cairo

Egypt

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Chicago

United States

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A 9-to-5 working day in Cairo does not overlap one in Chicago at all — somebody has to take the call outside office hours.

City	Time zone	UTC offset
Cairo · Egypt	Africa/Cairo	UTC+3
Chicago · United States	America/Chicago	UTC−5

What is the time difference between Cairo and Chicago?: Cairo is 8h ahead of Chicago. This gap is not fixed: daylight saving time moves it between +7h and +8h during the year.
Does the time difference between Cairo and Chicago change during the year?: Yes. Because of daylight saving time the gap moves between +7h and +8h.
When is the best time to call Chicago from Cairo?: There is no overlap between standard working days, so one side has to call early or late.