Sardinia driving

I'd at least inquire with the owner about the car deal she is proposing. First, I'd go to the AutoEurope website and see what results come up--you'll get a table showing various operators. But I don't put a lot of effort into understanding car insurance policies...
I'd look for a small SUV, something like an Opel.

Should we reserve a car from a US company before we go, or from an Italian company?

There's really no such thing as a "US company." There are large, multi-national car rental agencies, some of which are more familiar and better known to Americans. But even if they have a little "Avis" or "Hertz" sign on their counter, if you are in Italy you will be renting from a local company. The "American" company logo is just a plastic sign. Same as the plastic signs with the names of companies more familiar to Europeans (Auto Europe, Europecar, Sixt, etc.).

It's worth it shop around a bit (when comparing, be careful you are comparing apples-to-apples). Should you trust your Air BnB host for this? (Should you trust them for anything?). Hard to know that. At some point you have to trust something or someone. Personally, I've had plenty of good experiences with many different car rental companies (and, frankly, never had a particularly bad one, though others certainly have). How well do you/can you know your Air BnB host - or the car rental company?

what is a suggestion for best car for 4 adults, given relative comfort and road conditions?

"Road conditions" are just fine (as good as, in fact better than, most roads at home - at least compared to the roads where I live).

You want as small a car as you can manage (because this is Europe). But your biggest challenge may be getting a car large enough to hold 4 adults - and their luggage - comfortably. How much baggage will you have and how large will that be? Don't underestimate how much space you're going to need to be comfortable. That - smaller car for driving in old cities and villages, but large enough car for comfort - that's the balance you need to strike. It can be tricky (especially if you have large/multiple bags).

We were just in Italy last month and rented a Peugeot 5008 SUV. We were 3 people with 3 giant suitcases plus smaller stuff. It would have seated 4 adults comfortably and carried 4 smaller suitcases. Good luck with the trip!

Hertz or Avis in Italy are just franchises. That means it’s an Italian company that has entered a franchise agreement with Hertz or Avis.

We don’t know what kind of arrangement your host may get. It’s likely with a local rental outfit. Whether the price will be better or not we can’t know. Ask what the price would be then search and compare using for example www.autoeurope.com

4 adults with luggage need at least an intermediate size car. You might do with subcompact SUV like a Jeep Renegade or Fiat500X (both are made by Stellantis in Italy, are the same size since they share the same platform, and are very very common in Italy) but the cargo space is limited so you need all to have carry on size bags to fit all. A Jeep Compass (also made in Italy and popular) has a slightly larger cargo, but I think that might be intermediate. Opel Mokka (also Stellantis but made in Spain) is also popular but the same size as a Renegade and it is compact category.

Midsize cars are slightly bigger, like a Mini Countryman, or Toyota Yaris Cross (a crossover), or a Skoda Kamiq, or VW T Roc

If you have people who can’t travel with just a carry on (like my wife), then take a wagon. That would be the category of a Peugeot 308.

The Peugeot 5008 obviously will do, but that is in the large/SUV category in Italy so it will cost more.

If you need something larger than a 5008 then I can’t help you. Unfortunately they don’t sell the Canyonero in Italy.