I would take one night from Florence and add it to Rome, then rent a studio apartment in Rome to save money - not only on the room but on food as well.

I think you may find the flight to Sicily, may be two or three hours after your landing.

With almost one and a half months, one night at the airport will help you recover from the trip.

Just stay at the airport booking.com/FCO

Then you will find a 7 or 10am flight the next morning.

I didd not know you could stay at the airport! That has completely eased my fears! Thank you all so much! I have so much food for thought!
Thank you, thank you, thank you :D