Really my only comment is do do the the first 3 days in Rome to allow your body to adjust to the time change and because when you'll be arriving could be anywhere from early morning to mid-late afternoon depending upon routing. After 2 flights I wouldn't want to sent another ~90 minutes on a train to Naples along with going from airport to train station and waiting.
The train ride from Naples to Florence is less than 3 hours and would be backtracking but it's short and a nice ride.
Beyond that it sounds like a nice trip. Quick paced but not overly so.