Here is my biggest concern: have you considered taking the bus or train from Palermo to Agrigento and picking up a rental car there? There is good public transportation from Palermo to Agrigento. I think you may need actually two nights in Agrigento unless you plan to leave Palermo very early on that last day. You have to get your rental car, drive about two hours or longer to Agrigento, check in to your accomodation, have lunch at some point, then you will want to go to the ruins and probably the museum and allow yourself several hours for exploring those areas. Seems like a day packed with activity. Then you want to leave early the next morning for the Villa Roman del Casale. Give yourself two nights and you will have time to explore the city of Agrigento on your first afternoon or evening and use the next day to take your time to explore the ruins and museum. Maybe take a day away from Palermo.
I have used Palermo as a six night base before. Lots to see in Palermo and many daytrip options but some not so easy to get to by public transport. Possible daytrips from Palermo include: Mondello, Cefalù, Segesta, Erice, Castellammare del Golfo, Scopello, Borgo Parrini, The Sanctuary cave of Santa Rosalia on Monte Pellegrino, and many more. Without a car to do your daytrips I am not sure I would stay nine nights so I think it would be ok to take a night away from Palermo and add it to Agrigento. It is easy to get from Palermo airport by taxi, train, bus or shared taxi. There is an excellent youtube video here that shows you just how to do it. The bus and shared taxis are cheap and have a designated route and stops. They do not take you directly to your BnB or hotel. It is also easy to get to Monreale by public bus. I have never rented a car in Sicily but have used the trains, the buses and private drivers.
Siracusa/ Ortigia is a real gem. Stay In Ortigia. There are public bus routes that pick up at different points in Ortigia and go over to Siracusa and its archaeological park. Try to see the puppet show at the Teatro dei Pupi on Via della Giudecca. It was fun. Eat a sandwich or lunch at Caseficcio Borderi right next to the street market.
I like Catania. I just stayed there four days in March. As others have mentioned I too enjoyed the Military museum dedicated to the invasion of Sicily. Also check out the Palazzo Biscari which is about three blocks from the Sant' Agata cathedral. Lots of daytrips from Catania too: The Aci towns (Acireale, Acicastello, Acitrezza), Mt Etna, Taormina, Zafferana Etnea, the wineries, and more.