We will get a rental car in Palermo - which is the best company to rent a car from?
The one that offers a good price, and has a reasonable price for a level of insurance you are comfortable with. Be aware that many credit cards do not include Italy in their free coverage. I used Avis in Sicily (Had a company deal), but have used Europcar other places.
Should we get a rental car at the airport?
The airport is quite distance from town, to the West. If you are heading to Trapani, Marsala, or points West, then yeah. If heading South or to the East, then look within Palermo. Do consider where you really need a car, we started in Palermo, but only rented a car in Catania, to do South Central Sicily. Hard to say without knowing your itinerary.
If we have car problems while on the road, who do we call for help?
Talk to your rental agency, they usually have a number to call, you may also want to consider insurance that covers road service.
Is gas expensive?
By US standards? Yes. Figure about 7 euros per gallon, but then cars tend to have better mileage.
Anything else we need to be aware of while on the road through Sicily?
Cars tend to be manual transmission, but you can pay more for automatic. While not as bad as Tuscany, there are some ZTLs, Google it. Roads for the most part are pretty good, some of the speed limit signs are kind of confusing, but we had no issues.
If you get in smaller towns (and some large) roads can get narrow (like car width narrow) and very steep. Parking at a premium.
We had no real issues, and we found for the segment of the trip we did, car was the only logical option, But we used the car for a specific segment, picking it up after a week and three stops, then returning it before our last stop. We used the car for two stops over about a week and a half.