Again - you do not need a car to tour these places. All this can be done by train if you don't want to drive, easily and at less costs. Spend the extra money on a nice hotel instead, for example Gothisches Haus in Wernigerode. Wernigerode makes a convenient base because it is in the middle of the places mentioned and because it is the starting point of the narrow gauge railway up into the mountains. Besides, it is a gem of a place.

If you are set on Berlin, though, go straight there and spend the whole time there. 4 days is about minimum this city requires.