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

Yes. Because of daylight saving time the gap moves between −17h and −15h.

When is the best time to call Sydney from Chicago?

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

Chicago is 15h behind Sydney. This gap is not fixed: daylight saving time moves it between −17h and −15h during the year.

Chicago

United States

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Sydney

Australia

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

City	Time zone	UTC offset
Chicago · United States	America/Chicago	UTC−5
Sydney · Australia	Australia/Sydney	UTC+10

What is the time difference between Chicago and Sydney?: Chicago is 15h behind Sydney. This gap is not fixed: daylight saving time moves it between −17h and −15h during the year.
Does the time difference between Chicago and Sydney change during the year?: Yes. Because of daylight saving time the gap moves between −17h and −15h.
When is the best time to call Sydney from Chicago?: There is no overlap between standard working days, so one side has to call early or late.