I've been on this horse before, but I think 3 nights in Taormina is too much. From Catania, in the best of worlds, its an hour trip, but that's not allowing time for airport formalities (minimal) and getting to the transportation that will get you to Taormina. So yes, you'll be pretty tired upon arrival and if you need serious sleep before getting started, you may need three nights. I don't think Taormina, is worth more than a day so could you reduce your overnights to two there? I agree with others in that I would add a day to Siracusa, or to Palermo which would allow you to visit Cefalu, but it would be a shame to be in Palermo without going to Monreale, too. Each will eat up a half day.
Or, what if you cut out the overnights in Taormina entirely, and head directly to Siracusa from the airport? You could add those overnights to Siracusa (and/or Palermo). Then from Siracusa, do a day trip to Taormina. I didn't look anything up, but I'm pretty sure it exists. That would give you a few extra days to work with and you wouldn't be using time backtracking up to Taormina and back through Catania to Siracusa.
I'm with team husband on avoiding the one-night stays. Power sightseers can do Ragusa in a solid day, but you could stretch that a bit and day trip to Villa Romana del Casale from there, minimizing your one-night stays. Or with an early start and good driving luck you could start at Villa Romana, (which would be good as it is crowded in the afternoon), and go on to Agrigento. Valley of the Temples will take a few hours, but I'd allow a few hours for the museum as well. If you want to both, I don't think you could do all three comfortably in one day. So depending on your priorities, you'd need at least one night, and maybe two in Agrigento. After the temples and museum, there's really not that much to see in Agrigento that would fill two days. You may want to consider figuring out how to squish Agrigento's sights into one day and make that your one-night stop.
We spent considerable time in Trapani, and it's a nice little town, but there are no real "sights" to see. We used it as a base for a bunch of day trips and a few rest days. Is there something in particular that you want to see/do in Trapani or perhaps a specific day trip from there? Being a glass-half-empty person, I wouldn't leave the luggage in the car either, but squeezing in a maximum number of stops may require that you keep moving for a day or two and require that the luggage be kept in the car..