You don't need a car to access most of the major sites in Japan, including Nikko, because there is a well-developed system of trains and buses that will take you almost anywhere. For example, you can access Nikko by direct private railroad express train from Tobu Asakusa or, if you have a JR Pass, by taking the Tohoku Shinkansen to Utsunomiya and transferring to the Nikko Line. Once you're in the area, buses can transport you to spots such as Lake Chuzenji.

A further consideration is that Hakone is on the way between Tokyo and Kyoto, so it may make more sense if you have limited time, and the whole Hakone Free Pass circuit of trains, buses, funiculars, aerial tramways, fake pirate ships, and so on can be kind of fun and is very easy to follow.

As for seeing Mount Fuji, no matter how close you are, you have a better chance of actually seeing it in the winter than in the summer, due to the high humidity in the warmer months. I've mentioned this on Flyer Talk before, but one summer I visited some people who lived only 10 miles away, and the mountain had been invisible to them for weeks. On the other hand, the first time I ever saw it was from the roof of the library at my university in central Tokyo one crisp November day.