Can't remember which rental car agency it is (maybe hertz shinagawa) but one of them does have a black edition R35 atm. It's only for another month or so though and I believe it's booked solid. Nissan car rentals will likely have a 370Z as they always had the 350 and they have they skylines. R34 I suspect is waaaaaaaay too old to find for hire

Can't remember what the toll would be to yokohama (I assume daikoku futo?) but while it would be expensive by most international standards its not that much as its so close

For interesting roads it really depends how far you want to go. South west you have hakone/fuji area which also has a number of quite good car museums or to the north west you have places like the western end of saitama, gunma etc with some stunning mountain drives. In these cases however instead of getting a car in tokyo you're much better off catching the train out to the area you're travelling in then hiring the car from the station.