Hi there,
For a trip of this length I think you're looking at a 14 day pass to give you most peace of mind, although a 7 day pass from the 20th-26th could also well give you some savings. Lets take a look at your base JR ticket costs to get an idea of the overall costs of your trip for comparison, assuming you'll depart from Nartia within the 14 day period:
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Narita airport terminal 1 > Kyoto: ¥16330, (238mins, 592km)
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Kyoto > Nara: ¥690, (44mins, 41km)
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Nara > Kyoto: ¥690, (44mins, 41km)
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Kyoto > Hiroshima: ¥10990, (112mins, 380km)
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Hiroshima > Kyoto: ¥10990, (126mins, 380km)
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Kyoto > Osaka: ¥540, (28mins, 42km)
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Osaka > Kyoto: ¥540, (29mins, 42km)
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Kyoto > Odawara: ¥11950, (177mins, 429km)
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Odawara > Tokyo: ¥1450, (36mins, 83km)
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Tokyo > Nikko: ¥5630, (121mins, 150km)
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Nikko > Tokyo: ¥5630, (117mins, 150km)
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Tokyo > Narita airport terminal 1: ¥3140, (58mins, 79km)
JR ticket cost: ¥68570 for 12 legs
- 14 day rail pass (¥45,100) saving: ¥23470
Based on your JR ticket prices, you stand to save over ¥23,470 each with a 14 day JR Pass so I can certainly recommend this option over individual tickets.
The Green Car is certainly more spacious and exclusive, but is not available on all bullets between Tokyo and Kyoto. We have a good blog post on the benefits of the green car here.
From Kyoto, I think you've got all the bases covered - If you think you'll have time I recommend looking into trips to Koyasan on the Kii Penninsular to the south of Kyoto, and Kinosaki Onsen hotspring which is around 2 hours from Kyoto with a JR Pass on the Sea of Japan coast. Both are best as overnight stays, but could be reached in a day.
Hope this helps!