Naples is a wonderful and raw and is one of the oldest continuously inhabited urban areas in the world. A must see for anyone interested in history and true Italian life.

Naples and Bologna are my two favourite Italian cities.

Chiming in late, with a resounding YES for Naples, IMO!

You have what I would consider generous allotments of time elsewhere. I think you could cut Taormina by a day, Ragusa by 1 or 2, Siracusa by up to 3 or 4 (so a week, total, for Ragusa and Ortigia), and Palermo by 2 or thereabouts!

You'll have a great trip no matter what1