Hi there,
Here are the costs for your travel (assuming arrival at Tokyo Narita Airport):
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Narita airport terminal 1 > Tokyo: ¥3140, (62mins, 79km)
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Tokyo > Kamakura: ¥2100, (47mins, 51km)
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Kamakura > Tokyo: ¥890, (54mins, 53km)
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Tokyo > Nikko: ¥5630, (121mins, 150km)
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Nikko > Tokyo: ¥5630, (117mins, 150km)
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Tokyo > Kyoto: ¥13420, (164mins, 513km)
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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 > Miyajimaguchi: ¥400, (26mins, 21km)
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Miyajimaguchi > Tokyo: ¥17850, (331mins, 916km)
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Tokyo > Narita airport terminal 1: ¥3140, (58mins, 79km)
JR ticket cost: ¥64570 for 12 legs
- 14 day rail pass (¥45,100) saving: ¥19470
As you can see, you'll make good savings on travel with a 14 day Japan Rail Pass so I can certainly recommend this option as the "stress free" option.
7 day rail pass plus tickets
If you didn't mind altering your itinerary a little, you could move your Nikko trip to day 5, and then activate a 7 day rail pass from day 5-11. For the first 4 days, you could purchase tickets as needed, including the [Narita Express + Suica travel card deal], which will get you into Tokyo from Narita and also give you ¥1500 of free metro or JR travel. You may also like to look into the various Tokyo Metro 1 and 2 day travel passes which are also available to purchase at Narita Airport.
Booking online
Currently, you cannot book any of your trips online in English and can only make free seat reservations in Japan on presentation of your JR Pass. Despite this, I don't think you'll have any problems with booking a seat as there is normally plenty of availability. If you are concerned about this, I recommend you place all your reservations on arrival in Japan as soon as you have exchanged your JR Pass.
Hope this helps!