7 or 14 day Japan Rail Pass

Hi,

We have made the following initiary for our trip to Japan:

6 sept Arrival in Tokyo at 1.55pm, hotel in Tokyo
7 sept Tokyo, hotel in Tokyo
8 sept Tokyo, hotel in Tokyo
9 sept Nikko, hotel in Tokyo
10 sept Tokyo, hotel in Tokyo
11 sept Kamikochi (Japan Alps), hotel in Matsumoto
12 sept Hike Magome to Tsumago, hotel in Kyoto
13 sept Kyoto (Himeij castle?), hotel in Kyoto
14 sept Nara , hotel in Kyoto
15 sept Kyoto, hotel in Kyoto
16 sept Hiroshima en Miyajima, hotel in Kyoto
17 sept Kyoto, hotel in Kyoto
18 sept Mt Koya aka Koyasan, stay in temple in Koyasan
19 sept Mount Fuji, hotel in Tokyo
20 sept 6.20pm departure Tokyo

We have a few questions:
- Would you recommend us to go to Koyasan or cancel this and have one extra day to see Tokyo and surroundings? - Would you recommend us buying a 7 or 14 day Japan Rail Pass?

Many thanks in advance for your help!