You seem to have done your research. Logistically, Catania makes sense for the east coast, but I'd still prefer Siracusa because I place a pretty high value on seeing several of the small Baroque towns in the interior (Ragusa, Noto, Modica, Scicli) that are more easily visited from Siracusa than from Catania. Along with the medieval Ortygia district, the archaeological park and the archaeological museum, that tilts the balance toward Siracusa for me.
Catania does have an attractive historic district and a lively market (but there are two good ones in Palermo). Taormina is stunning but very touristy, so I'm not someone who would want to stay long there.
Sicily is an at-least-two-weeks destination, so you'll need to leave parts of the island untouched. Definitely go for nine nights rather than seven.
Do you plan to rent a car, or will you be using public transportation? I only used buses and trains and got around pretty well, but I had 18 days and could afford to be patient. There's not a great deal of train service (check trenitalia.com), and routes into the interior will often require buses. A key thing to know is that in many if not most cases, the buses do not run on Sundays. So be very careful about how you plan your itinerary for Saturday-Monday.
Do not under any circumstances attempt to drive in Palermo. A good mix of transportation methods might be to use the train along the east coast and between Palermo and Cefalu. In the middle of the trip, it would save you some time to have a car for the Baroque towns and the drive through the interior toward Palermo, though there is a rail line between Catania and Palermo.
It's interesting to see the interior of Sicily. A car might give you time to stop along the way in Piazza Armerina to see the town and the nearby Villa Romana del Casale. Perhaps you could also see Monreale (outside Palermo) before you drop the car off somewhere on the outskirts of Palermo. There is public bus service to Monreale, but on a short trip, the time saved by using a car already rented for other purposes could allow you to fit in one more sight.