Hi Morvegil,
Yes now is better your itinerary in terms of not making things so rush. I live in Palermo. I can tell you that traveling south Italy is kind of no stress when things do not work! Anything can make your trip slower .. things that you do not expect happens so be ready to be patient sometime ;) But is very worth for culture spots, people, food and money!
I can recommend as well to take one way flights, like fly into catania or palermo and fly back in the opposite way, so you will have more time.
Here some comments to your sicilia part:
May 8th:
Fly to Palermo
May 9th:
Site See in Palermo
Plenty of things to see.. but in case you don't know visit:
visit Santa Maria dello Spasimo (incredible feeling in a half destroyed church without the roof, where now classic and jazz concerts take place)- visit the palazzo branciforte asking to visit the 'Deposito dei pegni di santa rosalia' (google it)
May 10th:
Sicily (Rental Car)
Drive to Bar Vitelli (3 hours)
Driving to Bar vitelli from palermo.. you Must stop at Cefalu! you cannot pass by and don't see cefalu which is more worth then Taormina..
Drive to Taormina (45 mins)
May 11th:
Vist Mt.Etna
May 12th:
Drive to Syracuse (1.5 Hours)
May 13th:
Drive to Ragusa (1.5 Hours)
Stop in Noto for the unesco sites with barocco art..
Drive to Palermo (4 hours) - Plan back to Naples.
Don't drive back.. flights one way are cheap as well.. fly back from Catania.
May 14th:
Fly Home
I would suggest also to see the Aeolian Islands, which are fantastic! Stromboli island is unique in the world, and from there you can take a ship to naples, but you have no time unless you go to the islands instead of east sicily .. Palermo-cefalu-aeolian islands-naples... ships are twice per week.. so maybe this do not suite your trip.. but for next time spend some days in this amazing archipelago.
PS: Eat in Sicily:
in palermo a typical very affordable restaurant is calles Ferro di Cavallo.. while in the historical center find the famous 'Focacceria da Francesco' (which located in a lovely hidden square)
Wherever you can try this food: Arancine alla carne or al burro - Cannoli with fresch ricotta (buy them only if fresh (the best is when they fill this pastry in front of you) - in palermo the Panelle e crocche' is also a special local food, simple and tasty with some lemon .
PS2: I born in Naples, is so beautiful city, probably the most beautiful gulf in Italy, if you walk in naples by yourself is not dangerous at all if you avoid to show jewellery, photo machine and anything with value.. you can bring just consider that is easy to be stolen by a scooter if you hang your stuff on the street side.. just keep it not reacheble ;)
Enjoy your trip, will be fabulous :)