Totally agree with Rosalyn: Sicily is a minimum-2-week destination, and even with that much time, painful decisions have to be made. Two nights in Catania would be not much more than one full day by the time you settle in if you stop at the Villa Romana del Casale on the way across the island. That'll cover Catania's sights (and then some), but what about Taormina, Siracusa, Ragusa, Noto, Modica, and Scicli--the Baroque towns in that part of the island? Three nights in Palermo, if at the beginning of your trip when you're jetlagged, will be just two full days. That's minimally adequate for Palermo, I guess, if you don't want to see many of the beautiful churches and chapels. It probably won't allow time for Cefalu, because Monreale is a must.
I note that you plan to rent a car. That will not be a help while you are in Palermo. Simply crossing the street as a pedestrian is challenge enough in Palermo. Although there's bus service out to Monreale, a car would be more convenient, so I'd plan to pick up the car on the day I left Palermo and see Monreale before heading east.
Really, I wouldn't do what you're planning; it would be too frustrating. And it will require extra flights to get to and from Sicily. I hate having to fly in the middle of a trip; at least I'd want to have enough time to fully enjoy the destination.
Many of us combine Matera with Puglia. I'm afraid I don't know what the major destinations immediately around Matera might be. I think most folks like to have one full day in Matera, which usually means two nights. You can do a lot in Puglia in 4 or 5 days if you have a car. A lot of the towns are very small and can be seen in just a few hours. However, Puglia is a long, skinny region, so there's a lot of driving involved in hopping between places like Lecce, Gallipoli, Monopoli, Alberobello, etc. Lecce is a beautiful Baroque city with a lot of sightseeing possibilities and deserves at least one full day (it makes a lovely base), and I think Bari has a nice historic district.