Ok...there are more than a few subsections here.  RS forums have always provided real thoughtful travel info and I'm hoping this post garners lots of 'we've been there and this worked' comments! 
- My husband and I are each 68, live in NYC and have traveled a lot.  We can easily walk 3-5 miles a day - (we are city folk and deal with subways and stairs all the time) ...on a good day, 6.5 miles.
- We have 16 days on the ground (the first day for me is always a jumble of jet lag) May 20-depart June 5.  Flying into Palermo and out of Catania.
- We have LOTS of ideas but think we are going to end up following a pretty usual route...counterclockwise.  That said, I would prefer not to rent a car (and, trust me, I'm a great NYC driver as is my other half). I will if I have to.    I am considering a mix of public and private transportation.
- I prefer not to run around too much like a chicken without a head but understand that day-tripping or, for example, a stop in Agrigento prob. means 1 night stay occasionally.  My hope is to have 3 stops with 4-6 nights in one place and then day trip from there (and those become apartment stays so we have laundry).
- We will hire a private guide in a number of places (he's a ruins geek and has loved the ruins tours we've done in Italy and elsewhere).  I want to do an advanced cooking class but no clue where that wouldn't feel 'here's how to make pasta' - trust me, I already know!  But agriturismo tips are welcome.  
With all that in mind, here's my first go at dividing the time - questions and ideas/tips on any section PLEASE FORGIVE SPELLING ERRORS! We are open right now to adjusting this in any direction - the only thing in place right now are the plane tickets.
6 nights in Palermo: side trips can include Monreale Cathedral (no further than a subway ride for us - just like getting on the subway in Brooklyn and getting off on the Upper East Side! - so we see that as 'in Palermo'.)  Segesta, and maybe Selinunte and/or Erise?  are those worth it? 
       *** WOULD LOVE suggestions on whether it's worth it to stay in Trapani vs. visiting parts of it??? 
1 night (maybe 2 - seems unncessary) Agrigento. Museum and Valley of the Temple with guide.
4 nights in Ortigia - day trips to Noto and Modica - husband curious if we should stay there instead as opposed to day trip.
5 nights remain in this scenario and it could be in Catania with day trips to Etna (which is a have-to for us), Taormina - the theater ruins!!! and a walkabout). We have a 9:55am flight out of Catania and have to be there at 7am.
Unanswered for us are Ragusa, Villa Romana, a more inland agriturismo experience, and I was pumped to go to the Grotta del Genovese - Levanzo but that seems to be an impossible task in terms of time (we were just floored at the South of France cave drawings - unsure if it's just a teeny spot in this cave or extensive but I suspect this is for another time.)
Like everyone, we are looking for an experience that isn't super-touristy all the while going on a super-touristy route. We think this is our one trip to Sicily (Japan awaits before we turn 70) and want to really see it - it's not a week-on-the-beach vacation but we will find pockets of relaxation as we go. Hence the figuring out the puzzle of our stays.
THANK YOU SO MUCH should you choose to tackle parts of this.
bean