Hi there,
Yes, looking at your itinerary on Hyperdia train planner you'll save a lot by using the Japan Rail Pass. The main costs for your trip will be on the following legs based on starting your rail pass on day 3:
- Tokyo > Kyoto: ¥13,420
- Kyoto > Hiroshima: ¥10,990
- Hiroshima > Tokyo: ¥18,250
- Tokyo <> Nikko: ¥11,260 (return)
- Tokyo <> Hakone(odawara): ¥7680 (return)
So the calculation to make is:
Ticket prices (¥51,600) - 7 day Japan Rail Pass (¥28,300) = ¥23,300 saving with the JR Pass
(I didn't include minor travel like local trains to/from Nara and the ferry to/from Miyajima - they are free with the JR Pass).
About which day you plan to go to Kyoto, I'd probably do the same and go to Kyoto on Day 3 to maximise the value of the Rail Pass with your Hakone trip. You'll also need time to recover from the flight in, so waiting to Day 3 to begin your exploring makes a lot of sense.
Hope this helps!
M