Hi there!
For a 7 day JR Pass to be worthwhile you need to spend over ¥28,300 on JR tickets. For a 14 day pass you need to spend over ¥45,100 on JR tickets. As the total cost of your legs is ¥43,040 I think we have to look at the 7 day pass for you. Lets take a look at your JR ticket costs for the most expensive long distance travel:
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Tokyo > Nikko: ¥5,630 (121mins, 150km)
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Nikko > Kyoto: ¥17,710 (286mins, 663km)
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Kyoto > Tokyo: ¥13,420 (161mins, 513km)
JR ticket cost: ¥36,760 for 3 legs
As you can see you'll make good savings with a 7 day pass in addition to having the flexiblity to do any further JR travel during this time for free (eg, travel around Tokyo, or return to Hiroshima/Miyajima or Nara). I recommend you activate a 7 day pass form the 14-20th.
You should also look into either the Keisei Skyliner + 2 day unlimited metro pass or the [Narita Express + prepaid Suica card] for discounted travel to/from Narita and travel within Tokyo. I recommend the Skyliner deal as I think it's better value for money, but it will depend on your planned usage of the metro/JR lines within Tokyo. You may also like to purchase further 1 or 2 day unlimited metro cards - be aware that these special passes are only available at Narita!
The period you plan on travelling will be relatively quiet I think with no major public holidays during this period so I don't think you'll need to make any reservations. If you'd like to do so you should visit the JR ticket office once you are in Japan with your JR Pass as reservations are free for JR Pass holders.
Hope this helps!