Mid-September is still peak season in Greece and on most of the islands. Because of that, pre-booking a car is recommended, especially if you want an automatic, which isn't very common.
My method to find a rental company when visiting an island for the first time is pretty simple: I don't search the web. Instead, I use Google Maps, center the view on the port, and search for "Rent a car." From there I check out the websites of the listed local rental companies to compare their terms and prices. I also take a look at the reviews (though I always take those with a grain of salt).
My Golden Rule: I only choose local rental companies, and preferably ones that do not require prepayment, deposit, or a credit card number upfront. I look for companies where I can book via email or a contact form on their website, and then pay when I pick up the car, either by credit card or in cash (sometimes paying in cash gets you a discount).
When it comes to insurance, basic coverage is enough for me. Most of the time, they will offer add-on insurance options. To each their own on. Just keep in mind that even if you buy extra insurance, it usually still doesn't cover all damages, so make sure to read the terms and conditions carefully.
This approach always works for us. That said, I'm sure some people prefer a more by-the-book method with all kinds of official guarantees. That works too.
We usually only rent economy cars (Category A or B), meaning small or compact models. There's really no need for a high-end convertible, though that can be fun on the islands if you're into it.
Regarding parking, the only spots where driving or parking can be tricky are around the ports and the main towns (capitals). In Naxos, for example, Naxos Town is notoriously difficult to navigate and park in from June to September. It actually gets pretty chaotic around ferry arrivals and departures.
Everywhere Else, villages, beaches, or along the waterfront, you don't need to worry. You can park pretty much anywhere you want as long as it isn't explicitly prohibited. Just look at what everyone else is doing.
Driving on the islands is very straightforward with no major difficulties, provided you know more or less how to drive in rural areas. The most dangerous drivers on the road are usually other tourists :-)
Most road signs are written in both English and Greek.
Don't be surprised if a paved road suddenly turns into a dirt road. Other specialties in the islands:
The wind. Be careful when you open your car door, watch out for the wind gusts. A sudden gust can make you lose control of the door, and it could slam into something and cause some damage.
Goats: They wander on the roads and do not respect traffic rules :-)
I can't really speak for Crete—the island is just too big to generalize, and the last time I was there was over 30 years ago (plus, we rented scooters).
There are no ZTLs or other types of restricted zones in Greece.