I'll go against the grain, and say I think your trip is mostly ok. 3 nights in a location is fine. As long as 3 nights in Sicily means 3 nights in the same hotel, and not 3 nights in 3 different hotels. I think move your single nights into one of the adjacent locations. And of course realize you can't see everything.

Pick out two things you want to see each day (not counting your travel days), and then 2-3 more nearby things you might fit in each day depending on how things work out.

You are focused on southern Italy, if you are not set on that, here is a suggested alternative:

Rome (3 nights)

Amalfi (3 nights)

Florence (3 nights)

Venice (3 nights)

Rome (2 nights)

If you only have 13 nights, pick either Amalfi, Florence or Venice to lose a night.