I am on the fence about renting a car and trying to go from one country into another, especially if trying to drop it off in one we didn't start in.
Greg, no issue taking a car from one country to another. There is no passport control, and customs only ever do spot checks, 9/10 times you do not stop at the border.
But, there is a big issue trying to drop off a car in a different country to where you picked it up. This will either cost a lot or not be possible.
Geneva airport is on the border, so you have a choice of a Swiss car (for returning in Zürich) or a French car (for returning in France).
August/September is a bit close not to have booked your return flight.
If you want to drive on Swiss motorways/autoroutes/Autobahns you need to buy a toll "vignette" sticker which costs CHF 40. A Swiss car will already have one.
If you get as far south as Avignon, you could return from Marseille airport.
My suggestion for a 7-day tour: From Chamonix east through the Mont Blanc tunnel via Aosta (Italy) to Stresa on Lago Maggiore or somewhere on Lake Como. North over the Simplon pass to Zermatt, or somewhere smaller in the same valley. Then cut back to Geneva or Zürich for the flight home.