7 days in Sicily and driving between cities

You can do it without a car. You have barely time to see Palermo and Siracusa. You can take the Interbus bus from Palermo to Siracusa. Actually I would probably eliminate one night in Catania and add it to Siracusa.

Cutting through the center of the island is about a 4 hours drive between Palermo and Ortigia without any stops. The drive will be longer if taking the coastal route. By taking precautions one can reduce the likelihood of being broken into, but there are no guarantees. When we drove around the island we didn’t have any issues.

Keep all luggage out of site
Have nothing in view; coins, maps, bags, phone cables, etc.
Hide everything prior to pulling into the parking space so the trunk or hatchback don’t need to be opened after parking. Opening the trunk allows prying eyes to know there are things in it,
Don’t park in remote locations where there are few people passing by.
If parking in a parking garage, or anywhere, try and park where people frequently pass by.
If possible, back into the parking place preferably backing up to a wall, or trees, etc., and close enough to the wall or trees where nobody can fit between them and the car. Also, close enough where a hatchback couldn’t be opened.

It's such an odd one, I'm curious what led you to choose this itinerary?

You're not doing any actual sightseeing so the car seems like an expensive hassle. And as Roberto mentioned, the bus makes a lot of sense. It's simple, comfortable and cheap.

You probably do not need a car at all. Yes, I've hired drivers for place to place transfers in Sicily; if you do not want to take public transport, that's a fine option. Besides the chance of theft, parking would be a concern, although I guess if you stick to this plan, the only place you would have to park would be Ortigia. You could pick up the car upon leaving Palermo, and turn in the car at Catania airport before taking taxi into there city for your stay there.

If this were my trip and I was sticking to that agenda, I might get a drive to take me from Palermo to Ortigia, making a stop for lunch or sightseeing if you want....we've done this a few times with the lunch stop and it worked out well. To find a driver, your first step might be to ask your hotel; we've always had good luck doing that. You can then compare their price with prices of NCC drivers you find online....

But curious as to why you chose those three destinations...is a visit to Etna on the agenda?

What places that you wanted to visit did you rule out because you thought that t hey would be "shut down??" You might be able to reconsider those...(??). Where would the idea of being shuttered even apply, other than coastal resorts that are not on your radar in any case? Towns like Ragusa, Modica, Trapani, Taormina--all those are cities with large local populations that are not totally dependent on tourism and would not be shut down" in winter. Winter might be the best time to see some Sicilian destinations that are jammed with tourists in warmer months. So while Marina di Ragusa and Marina di Modica might be quiet in February (but not "shut down" because many people live there all year round, the towns of Modica and Ragusa would be lovely at that time of year..)

If budget is not a huge concern, a driver could take you from Palermo to Siracusa/Ortigia, and then--later in the week-- to Catania. We've had great experiences using private drivers, although it does cost a lot more than taking the train or bus..you can make a lunch stop along the way if you like.... (I had one of my best meals in Sicily at lunch between Trapani and Agrigento, while the driver waited to guard the luggage in the car..)

None of this will likely change your plans but you could keep it in mind for next time.

My wife and I were there for about 12 days, so we had more time than you did.
But .... my two bits worth is that there are GREAT sights to see in the south and middle of Sicily.

The BEST Greek temples > Agrigento
The BEST Roman mosaics > Villa Romana del Casale
Wonderful hill cities > Ragusa, Modica

All of these require a car. They are a few hours apart from each other, so they REQUIRE TIME, which you may not have.

Ragusa and Modica are viable day trips from Siracusa by train and/or bus, though I wish I had spent at least a couple of nights in Ragusa.