Car rental company at PMO

You can rent from almost all companies, but you'd read exactly and understand their T+C. And if you take their offered insurance, without any deduct, no company will have an interest for finding any damage.
You know, you'll need an IDP. And you should inform yourself about all the different traffic-rules and laws in Italy, esp what "ZTL" means. Any ticket in Italy is very expensive.

certain companies charging for alleged damage

All of them try to do it, at least up to the 1000€ deductible amount the insurance coverage generally entails. It's extra revenue since they will never fix a small dent or scratch but they for sure will charge your credit card for it.

The way to get around that is to have full zero deductible coverage (www.autoeurope.com, a broker that works with major rental car companies, offers that option for about $6-$10 a day extra). In your case it would be an extra $30-$50 (depending on time of the year) for the 5 days.

The other option is to make sure you use a credit card that will reimburse you for the payment of the non covered damage when you return home. It requires some extra paperwork on your part, if you have to pay for damage, but if you want to save the $6-10 a day extra for a zero deductible coverage option, it's a viable option. Check with your credit card before leaving if your credit card offers that coverage, not all cards offer that option in Italy.

Go to www.autoeurope.com
select from (pick up) PALERMO ALL LOCATIONS to (drop off) PALERMO ALL LOCATIONS
enter the dates and times you want.
Look at the deals available.

When you find the deal you want, call the 1-800 number and mention to the agent the quote you have online and ensure you want the coverage you need.
I checked for 5 random days in May this year, and the best deal, in my opinion, would be an SUV/Wagon with AVIS for $225 for the entire 5 days for an automatic car (same price for manual cars, if you prefer manual). The site no longer gives the option to choose the type of coverage, so when you call the agent make sure you are covered with the insurance policy you want (deductible or zero deductible). Deals vary continually, so be aware of that.

Additionally, coverages such as windshield damage and road hazard/tire damage are oftentimes additional coverages and not included even in the zero deductible CDW coverage. So, as stated previously, ensure you understand exactly what is covered with the insurance you elect.

Ok.

Since the Autoeurope's website has changed a bit and I didn't see the choice of coverage, I tested on www.autoeurope.com for a car pick up in Rome Tiburtina on June 22 and return on July 28 at Fiumicino (so over a month), just to try.

The car is $1595 for the entire 5 weeks' period (Hybrid Renault Symbioz SUV or similar)

Once you select the car deal, and the exact pick up and location, the coverage options come up.

There are basically 3 coverage options:

BASIC PLUS (zero extra cost).
This option covers only
-Liability,
-Fire,
-Collision Damage Waiver (CDW), and
-Theft protection (TP)

PARTIAL PROTECTION ($3.66/day extra)
In addition to the Basic Plus coverage items, this option gives also a Refundable Deductible (aka refundable excess)

FULL PROTECTION ($10.88/day extra)
This option covers everything included in the Partial Protection, but also adds the following items:
-Wheels, Glass, Roof, and Undercarriage Damage Coverage
-Windscreen, Mirrors, External Lights and Lamps
-Roadside Repairs and Towing costs
-Key Cover and Lock out

If you try Palermo for your dates the insurance options are likely to be similar at similar (or maybe the exact same) price.