Return to Palermo’s airport for the car hire – Avoid driving in Palermo itself. Palermo’s airport is located west of the city on your way to Trapani.
Begin exploring Erice – You can see Trapani in the morning and at night after day-tripping.
You would only require half a day for Segesta – You might combine Erice and Segesta in one day, but you could spend a full day in Erice.
Caltagirone would serve as a convenient base for exploring Piazza Armerina, Ragusa, and Modica.
Do not base in Catania if you plan to hold the car hire - Base outside Catania if holding the car!
Lot of questions on Catania:
Reconsider staying in Catania - Consider dividing your time between Taormina and Siracusa in Eastern Sicily. You can easily day trip to Catania from Taormina or Siracusa. See Etna from Taormina.
What to cut?? Your itinerary is perfectly acceptable as is with the exception of basing in Catania with a car, but you might consider cutting Cefalù so you would enjoy more time in Palermo.
You could also cut a night from Agrigento – One night would suffice in Agrigento.
Store your luggage at your hotel and continue exploring Agrigento on the morning following your 1-night stay, but then move onward for that night to better position for a full day of sightseeing at your next destination.
You could then spend that night in Piazza Armerina. Store your luggage at the hotel and see the Villa Romana del Casale the following morning, and then even move onward to a night in Ragusa or Modica, especially since these towns are best enjoyed by living and experiencing more than day tripping.
If you were to spend a night in Ragusa or Modica, you could then drive onward to Siracusa and then end in Taormina.
Or continue with a base in Caltagirone, but allocate another day for Siracusa.