I spent over two weeks in Sicily early last June and loved it. I'm a bit older than you and share your appreciation for simpler accommodations. I used venere.com for most of my Italian bookings (read the reviews), and that worked better for me than showing up at the local tourist office. Be aware that the Palermo office, while helpful, will not book a room for you. Be sure everywhere you stay has air conditioning. I found lodging costs very reasonable, and the food is great.
I don't recommend the night train from Rome. I'd investigate flying if I didn't have time for day trains.
I can only handle a few ancient sites on any one vacation, so I skipped a couple of spots that would be important to you, Segesta and Agrigento. I enjoyed the Greek theatre in Siracusa (but stay on the historic Ortygia peninsula). There's also an archaeological museum in Siracusa. As is common, I believe that museum isn't air conditioned.
The Villa Romana del Casale outside Piazza Armerina has wonderful mosaics. The site can be reached by public transportation, I believe, but I didn't time it right and took a taxi to and from Piazza Armerina (arrange the return in advance). One of my guidebooks mentioned a significant Greek site not too far away that was said to be lightly visited. I think it's called Morgentina. I don't know how accessible it is via public transportation.
Other places to consider are Taormina (fabulous site and very picturesque, but prepare yourself mentally for hordes of tourists); the inland towns of Ragusa, Noto, and Modica (public transp. from Siracusa); Cefalu (picturesque town with high concentration of tourists); Palermo (sort of rough-and-ready Naples-like experience but fabulous Baroque churches and oratorios, including the incredible Monreale); and Trapani (attractive west-coast town in its own right but also a good base for a day trip up to the evocative Erice, which I'd call a can't-miss for anyone visiting western Sicily).
I enjoyed Catania, but it's a big-city experience without quite the density of sights you'll find in Palermo. On a trip of less than two weeks I'd probably skip it, but it has an attractive historic core, and I always appreciate a real city that's obviously not being run for foreign tourists.
Inter-city buses are rare things on Sundays (and I assume holidays as well), so you'll need to plan your transportation with some care if you are not going to rent a car. Having a car would allow you to cover a lot more ground in the interior. An air-conditioned car would be a refuge from the (expected) heat and an improvement over hanging around outdoor bus stops. I didn't drive in Sicily myself. The buses were reliable, as we're the (limited) trains, except for a lengthy delay at the Straits of Messina due to track work.
