Hi there!
Assuming you start your potential 14 day travel period on the 4th, your costs will look like:
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Tokyo > Nikko: ¥5,230 (121mins, 150km)
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Nikko > Tokyo: ¥5,230 (121mins, 150km)
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Tokyo > Takayama: ¥14,300 (273mins, 532km)
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Takayama > Kanazawa: ¥4,180 (161mins, 148km)
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Kanazawa > Osaka: ¥7,440 (154mins, 267km)
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Osaka > Hiroshima: ¥10,220 (109mins, 345km)
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Hiroshima > Kyoto: ¥10,590 (112mins, 380km)
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Kyoto > Nara: ¥690 (45mins, 41km)
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Nara > Kyoto: ¥690 (45mins, 41km)
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Kyoto > Osaka: ¥540 (28mins, 42km)
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Osaka > Kansai airport: ¥3,340 (57mins, 64km)
JR ticket cost: ¥62,450 for 11 legs
Based on your travel listed you'll make great savings with a 14 day pass so I can certainly recommend this option. Note that for Mt Koya the JR Pass will get you to Osaka where you'll want to use the (non-JR) Nankai railways up to Koyasan.
As far as suggestions go, I think the route to Takayama via Nagoya + WideView train is really scenic. I think out of Nara and Osaka on the 11th I'd plan on staying in Osaka and visiting Nara perhaps later while you are in Kyoto as a day trip. Nara doesn't have a huge amount of night life, and it's away from the main shinkansen lines that you'll need the next day to get down to Hiroshima.
As a final recommendation, don't forget to take advantage of either the [Narita Express + SUICA deal] or Keisei Skyliners + 2 day metro pass deals to get you into Tokyo from Narita with some free travel around Tokyo built in. You can buy these tickets on arrival at Narita.
Hope this helps!
