First trip to Japan / 9 full days

We are a couple going to Japan for our first time. We want to stay in Tokyo for 3 nights, and take a day trip to Nikko. We're talking about buying a 7 days JR Rail Pass, travelling to Kyoto and Hiroshima. Can we do that in just 9 days ? Is it possible to include Mt. Fuji on the way to Kyoto ? Which route do you recommend ? Arrival Tokyo February 12, in the morning. Departing February 21 in the morning...
Thank you very much. Jesper, Denmark

Can we do that in just 9 days ?

Yes.

Is it possible to include Mt. Fuji on the way to Kyoto ? Which route do you recommend ?

Yes. The route depends on where you want to see Mount Fuji. In the extreme case, you do not even have to get off the train, as the train between Tokyo and Kyoto passes near the base of Mount Fuji, and offers nice views of the mountain on days with clear weather.

For a closer look, I recommend the Fuji Five Lake region. You could incorporate it as follows: Tokyo - Fuji Five Lakes - Kyoto - Tokyo

About the Fuji Five Lakes:
http://www.japan-guide.com/e/e6900.html

How to travel between Tokyo and Fuji Five Lakes:
http://www.japan-guide.com/e/e6905.html

How to travel between Fuji Five Lakes and Kyoto:
http://www.japan-guide.com/e/e6905.html?aFROM=2158_6900

Alternatively, you could visit Hakone instead of Fuji Five Lakes, but note that the chance of seeing the mountain is a little bit lower.

About Hakone:
http://www.japan-guide.com/e/e5200.html

How to travel between Tokyo and Hakone:
http://www.japan-guide.com/e/e5206.html

How to travel between Hakone and Kyoto:
http://www.japan-guide.com/e/e5206.html?aFROM=2158_5200