Hi there!
Given that you're in Japan for 17 days, I think your key options are either a 7 day pass (¥28,300), or a 14 day rail pass (¥45,100). A 7 day rail pass will require pretty drastic changes to your itinerary, whereas a 14 day rail pass will allow you to do all of your JR travel from Day 4 onwards. To get an idea of which is better for you, lets first get a breakdown of your costs:
- Tokyo > Nikko: ¥5430, (121mins, 150km)
- Nikko > Tokyo: ¥5430, (117mins, 150km)
- Tokyo > Kamakura: ¥1900, (48mins, 51km)
- Kamakura > Tokyo: ¥890, (58mins, 53km)
- Tokyo > Kyoto: ¥13220, (164mins, 513km)
- Kyoto > Nara: ¥690, (45mins, 41km)
- Nara > Kyoto: ¥690, (44mins, 41km)
- Kyoto > Hiroshima: ¥10790, (112mins, 380km)
- Hiroshima > Tokyo: ¥18050, (292mins, 894km)
- Tokyo > Narita airport terminal 1: ¥2940, (58mins, 79km)
TOTAL: ¥60030 for 10 legs
As you can see, a 14 day pass will give you minimum savings of ¥14,930 on your JR travel, with the freedom to use the rail pass more as you are in Tokyo (map of JR lines in Tokyo for reference).
I think rather than compressing your travel up into 7 days, the 14 day pass will be a good option as it will allow you to spend time outside Tokyo and stil make great savings on your travel without having you spend 10 days in Tokyo.
As a thought on your time from Day 10-14, a day trip to Kurokawa Onsen from Hiroshima will probably be too much to fit in a day without an overnight stay (and you really won't get the most out of it). You also may struggle to spend 4 nights in Hiroshima. I think you could reduce your stay in Hiroshima by 2 nights and spend them on Kurokawa / Aso, or another location like Naoshima, Koyasan, Fukuoka or Kinosaki Onsen.
For Day 1 - Day 3, you should make sure you take advantage of either the Narita Express and Suica metro card, or the Keisei Skyliner + metro travel card deals that will give you some savings for your initial few days.
I think your itinerary looks great! Have a good trip!
