Sicily

Sure, that works. You won't need the car for Syracuse unless you are planning day trips, and do not want it in Palermo at all.

Some might say 4 days in Syracuse is too much, just for Syracuse/Ortigia, but then I'll be spending 4 nights there in April, and I have been there before. If not a day trip, you might consider a night or two from Syracuse for Ragusa/Modica on the way to Agrigento, but just a thought, do your own thing.

I would not eliminate Palermo. It's a gem and well worth 3-4 nights on this itinerary

All in all, it's a pretty good plan IMO. You could shave a day from Siracusa and Palermo and spend 2 nights in the Baroque Hill Towns rather than daytripping. I think you'd have a richer experience and less autostrada time.

Trapani parking can be a bit of a hunt. Ask your hotel for advice and use the EasyPark app.

Syracusae is one of our favorite places in Sicily. I hope you will have a cae and you can use it as a base to see Notto, Rugusa Ibla. We spent 3 nights in Trapani a couple of years ago. We went to Agrigento on our way to Origia/Syracuse. It worked out well for us.
Just before you go check what is going on in Sicily now. In view of Cyclone Harry, which hit Sicily, Calabria, and Sardinia badly a few weeks ago. There is a state of emergency there now.
We were planning to go to Sardinia in September and will probably change our plans since there are a lot of distractions in the area..

Taking into account suggestions received we have switched up a bit.
Flying in to Catania from Malta early am. pick up rental car

4 nights Ortigia- day trips to Noto, Ragusa, Villa Romana del Casale, time around Ortigia/Siracusa

1 night Agrigento Valley of the Temples

3 nights Trapani - day trips Segesta, Erice, Selinunte, Salt Pans Moria. Egadi Islands

3 nights Palermo we will return the car before staying in Palermo. Bus or train day trip to Monreale. Then either Cefalu or San Vito Capo.

My question is should we stay the 3 nights in Trapani or stay in Castellamare del Golfo or a mix of the 2

We are travelling September 28 to Oct 9.

Very much appreciate any help.

Ive not stayed in Castellamare del Golfo. It's pretty and a more resort-y vibe than Trapani, but ultimately isn't it a lot more inconvenient for your desired day trips?

I've stayed in both Trapani and Castellamare del Golfo. I would suggest Trapani for your purposes.