Sicily in 6 days

I think your plan has you covering too much in a really tight timeframe - essentially three different regions which are not close together. Pick one region, else you'll be driving a lot and spreading yourself too thin. Six days (even full days) is really not a lot, so you'll arguably get a more relaxing, higher quality trip if you concentrate on cutting some of the geographical distances (there is much to do in every region, so you can save the other (s) for another trip). It sounds like the east location is a given, since I assume you have no option but to fly out of Catania to Naples. So just do a round trip to/from Catania. You can pick 2 bases, one for the northern area around Mt. Etna and Giardini-Naxos/ Taormina and one to the south (Siracusa makes a perfect base). In 6 days, you can see: Taormina, Mt. Etna, Siracusa, Noto, Ragusa, and perhaps Modica too (the last three are very close together). That would still be an ambitious 6 days, but with a car, could be feasible.

This is a good website (just ignore all the ads) to get you familiar with the areas:
http://www.bestofsicily.com/roadmap.htm

If you want to sample both, you could do 2-2-2 nights, staying in the west, then central, then east. It would be very fast paced and you may not be able to cover everything you've listed. More of a sampler.

If you think you may go back, I'd choose 2 locations and go back for more on another trip.

If you do 2 or 3 nights on the east, and I'm sensing you really want to go to Etna, with a car you could base in or near Catania and see Etna one day and Siracusa the next. The distances are small and the driving between towns in the east is not difficult.

Also, if you fly out of Catania and get lucky with your takeoff direction, you may get a front row seat to Etna.

Another way to get a great view of Etna, without taking a day to go there, is to get a hotel with a view of it. We stayed in a villa in Giardini Naxos whose balconies all faced Etna. Our evening picnic dinners on the balcony were unforgettable.

I would agree with upthread suggestions for sticking to a region that is accessible by train or automobile. We took a tour after southern Italy that included Mt. Etna & ended in Taormina. On our own we took the train to Siracusa and spent 5 nights in Ortigia. We did a day trip tour to Noto, Ragusa and Modica. Another option would be eliminate a night or two in Ortigia and pick one of the Barogue towns to day trip from. We took the train to Catania and spent the day and flew home the following morning. If time is limited I think "less is more." Next time we will devote more time to see the west side and revisit the east! The end of September should be lovely!

Having just spent 12 days in Sicily in April, and planning a return visit (with a friend who was not with me earlier) for 7 nights in a few months, this is my suggestion: If you have only 6 nights, and can move really quickly, you could:
Day 1: land Palermo. Sleep Palermo
Day 2: Palermo; afternoon to Monreale (I'd leave out Cefalu). Sleep Palermo.
Day 3: day trip to Erice/Trapani. Sleep Palermo
Day 4: leave early. Drive to Agrigento (2 hours), spend day; drive to Catania or Siracusa (3-4 hours). Sleep Cat or Siracusa (I believe it's about a 2 hour drive from Ortigia/Siracusa to Mt. Etna, and Ortigia/Siracusa is beautiful! Great east side base!
Day 5: see Siracusa/Ortigia. Sleep Cat. or Sir.
Day 6: day trip to Mt. Etna. Sleep Cat. or Sir.
Fly out of Catania.
This would therefore be 3 nights in Palermo, and 3 nights in either Ortigia/Siracusa or Catania.
Keep an eye on what's happening on Mt. Etna, and what might be closed. It erupted (let flow some lava) a few days ago.