Hi there,
As a 14 day rail pass is ¥45,100 and a 7 day rail pass is ¥28,300, lets first take a look at the overall ticket costs of your travel to get an idea of where you can save costs:
- Tokyo > Nikko: ¥5430, (121mins, 150km)
- Nikko > Tokyo: ¥5430, (117mins, 150km)
- Tokyo > Otsuki: ¥2860, (85mins, 87km)
- Otsuki > Takayama: ¥14820, (365mins, 580km)
- Takayama > Kanazawa: ¥4180, (186mins, 148km)
- Kanazawa > Hiroshima: ¥15250, (265mins, 605km)
- Hiroshima > Kyoto: ¥10790, (126mins, 380km)
- Kyoto > Osaka: ¥1680, (27mins, 42km)
- Osaka > Kyoto: ¥540, (29mins, 42km)
- Kyoto > Himeji: ¥5130, (68mins, 130km)
- Himeji > Kyoto: ¥2210, (91mins, 130km)
- Kyoto > Nara: ¥690, (44mins, 41km)
- Nara > Kyoto: ¥690, (48mins, 41km)
- Kyoto > Tokyo: ¥13220, (161mins, 513km)
TOTAL: ¥82920 for 14 legs
As you can see you'll make good savings with the 7 or 14 (or even 21 day pass at ¥57,700!). I think for this trip I'd aim to use the 14 day pass as you have a lot of travel planned, starting your travel with a 3 day stay in Tokyo to adjust to the country and get used to the timezone. From day 4 until day 17 aim to travel around Japan pretty much in the order you mention, spending a decent amount of time in each location and finally banking on an overnight in Tokyo before you head back to the Airport. As your savings are so large you could even return to Tokyo earlier if you wanted more time in the city before you head back.
Timewise, for each of your destinations it very much depends on your interests but I recommend overnighting on Miyajima (maybe 2?) and a visit to an Onsen town like Kinosaki Onsen. With so much time you could even consider going down into Kyushu where you'll find a semi-tropical climate, hiking around active volcanos, hot sand baths and so on.
I have assumed for Mt.Fuji you'll be going to Kawaguchiko (rather than Hakone) - Otsuki is the closest JR stop to Kawaguchiko, from there you'll need to take the non-JR Fujikyu line up to Kawaguchiko. I also left out the costs to and from Haneda as they are minimal and as they will be at the start and end of your 18 day period I don't think it makes to include them.
Hope this helps, if you have any more questions please ask!
