Renting a car in Chicago with California license on a tourist visa

Technically your license would have become invalid once you were no longer a permanent resident of the US. A "tourist" visa is NOT suitable for obtaining/holding a drivers license in California as only residents (temporary or permanent) are allowed to do so.

California drivers licenses issued to non-permanent residents should have an expiry no later than the persons legal status in the US expires, which basically means that the license can't expire any later than the date on your I-94 at the time you applied for/renewed the license.

Odds are that you will be able to rent a car using your license, and odds are that you will not have any issues as a result - but legally it's at best a grey issue. Presuming that you have a French license you will most likely be able to use that to both rent and drive a rental car, although given that your license will not be in English it's generally recommend that you get an International Driving Permit to make things easier.

Driver's licenses issued by one U.S. state or territory are valid in all of the other states and territories. In fact, it a constitutionally-based guarantee, deriving from Article 4, Section 1. Note however that if you have a valid driving permit in France, you will be able to use that to rent as well.

You will be asked for your current address when renting, but I would not expect any problems. It is of course quite common for people to fly to other states and rent cars there for both work or pleasure. When I first moved to Maryland I often rented at an airport in Virginia with a California license, and was never once asked any questions as to why I was living in one place, licensed in another, and renting from a third.

Your tourist visa status has no bearing on the validity of your driver's license provided that you are driving yourself or companions. Running a taxi service or being paid to courier is, of course, another matter entirely.

Regarding insurance, if you have personal auto insurance at home, first check your policy to see if it will cover your rental. Second, your credit card may offer some coverage, so review their offerings carefully— rental insurance provided by credit card issuers is very limited.

Credit card insurance is typically secondary insurance— it only steps in if you decline the CDW or LDW (collision damage waiver or loss damage waiver), and after all your other insurance has been exhausted. But most importantly, it does not cover liability insurance (if you get sued for crashing or otherwise doing something bad with the car) or personal injury insurance (for paying medical bills in the event of injury). I have never heard of any credit card that does. There is also uninsured motorist insurance, which covers you if you are in an incident and the other driver does not have the proper insurance.

The value of the various kinds of insurance are debated endlessly, so I will only say that for myself, because I do not have a personal auto insurance policy, I always get the supplemental liability coverage (it is "supplemental" if you have existing liability insurance, it becomes primary if you don't as in my case) and decline the rest. America is a litigious place.