Nancy, we booked all the way thru to Catania (round trip) on Delta. The last leg was ITA. We thought the ITA flight was fine. Would the leg to Palermo be on a ticket connected to the flight from Minneapolis? If so, I'd do ITA, knowing they'd be responsible to get you to Palermo. We have found the Rome airport very efficient/organized. We did carry on so I can't speak to checking/rechecking bags.
I can't recall if you've been to Catania. We really liked the cathedral and the WWII museum. We also did the RS walking tour. I wouldn't recommend more than a day or two because other cities in Sicily are nicer, but it was enjoyable for a couple days.
There is a bus from the Catania airport to Siracusa. If we had to do over, we would have gone directly to Siracusa and done just a day or two in Catania at the end of the trip.