I'd suggest cutting out the extremeties. Rome you can do by Ryanair easily and a car is a liability in the City. same applies to Florence, where you need at least two days.
Amsterdam I'd cut out too and do that on a Ryanair or whoever.
Pammy is right, in my view, about too tight a schedule.
my suggestion would be to scoot through Belgium towards Germany and pick up the Moselle in Trier. Drive along that for half a day to the Rhine and pick up the Autobahn. Between Stuttgart and Zurich there is the Rhine Falls Rheinfallen, at Schaffhausen, well worth a cup of tea break.
In Germany stay in Pensions, they are clean, simple and very cheap. Along the Moselle , you will find them for 40 to 70 Euro a room for the night.