This sounds like something I'd do, except there's no way I'd stay at an F1! Davos is pretty cool, but it is an awfully long drive as mentioned. Colmar is worth the visit as is the car museum in Mulhouse. But if you are really into cars and other types of motorized transport, I would say the number one destination for you should be the Chateau de Savigny-Les-Beaune. It's the most amazing museum I've been into in the world for that sort of thing. Not just for cars though, also motorcycles, fighter jets, tractors, fire engines, etc. Also small scale models of just about every motorized vehicle made.
You could stop there after Mulhouse on your way to Paris. But maybe allow several hours if this is your thing.
As for taking a car rental from France to Italy, I've done it countless times with Avis and Hertz. I think in the past there was an issue that Amex would not insure your rental for Italy, but that's no longer the case. At least not for me. Maybe it's because anti-theft control is much better in modern cars.