Hakone is an entire region, it's not a single city. You need to decide in what part of Hakone you want to stay. I'd recommend Gora. Hakone-Yumoto is more quickly accessible but less scenic.
With only one night in Hakone, you should spend the entire day 5 taking all the assorted transport systems around the area. Hakone and Kamakura in one day is way too much. Having been to Kyoto, I'd say you can skip Kamakura.
There's also plenty to see in and around Tokyo without making a day trip to Nikko, but you don't have to decide about that until the last minute anyhow. It takes about two hours each way to get to Nikko from Tokyo.