Just a few thoughts on the above:
Skiing country to country (i.e Cervinia to Zermatt or vice versa), there are no checks.
Most of the time, I have had no issue driving in the Schengen zone with border stops. However, I have been stopped crossing from Switzerland to France, but most of the time been waived through with just showing a passport.
My biggest issue is getting car tags correct for tolls:
France has the traditional toll plazas, so it's no issue.
However, last year I flew to Milan and was stopped at the Swiss border via Gotthard Tunel and forced to pay a Swiss $40 tag (vignette) fee. Not sure if that was to also use the tunnel.
Same thing in Austria if you are coming from Switzerland, you need to pull over and buy a tag at a petrol station.
Just FYI there is a $70+ roundtrip fee to use the Mont Blanc tunnel between Chamonix and Courmayeur.