07/07; Odawara to Hiroshima 10/07; Hiroshima to Kyoto 11/07; Day Trip to Nagoya 12/07; Day Trip to Himeji
Your currently planned course is
Shinjuku -- Odawara -- (Nagoya) -- Hiroshima + Miyajima -- (Himeji) -- Kyoto -- Nagoya -- Kyoto -- Himeji -- Kyoto.
(It may take more than 4 hours to go by Shinkansen Hikari or Kodama from Odawara to Hiroshima.)
Unless you hope to be at a specific site on a specific day,
you don't need to take such a zigzag course like this;
you can visit Nagoya on your way from Odawara to Hiroshima and Himeji on your way from Miyajima back to Kyoto.
Also, you have another option than a 7-day Japan Rail Pass:
a combination of Odakyu and JR regular tickets to go until Kyoto and a JR-West (5-day) Kansai-Hiroshima Area Pass for trips in western Japan.
(Any amount of transportation cost which is shown below is that for one adult passenger.)
The Odakyu regular cost to travel by Romancecar
(with a PASMO or Suica card + a Limited Express ticket)
from Shinjuku to Odawara is 1764 yen
(= 874 yen as a basic fare + 890 yen as a Limited Express Romancecar surcharge).
The JR regular cost to travel (in mid-July) by Shinkansen Hikari or Kodama
from Odawara to Kyoto with a stop-over at Nagoya is 13130 yen
(= 7020 yen as a basic fare + (3860 + 2250) yen as Shinkansen Hikari/Kodama surcharges),
on the condition that your seat is reserved both until and beyond Nagoya.
This JR basic-fare ticket is good for 4 calendar days.
You can start from Shinjuku, make stop-overs at Odawara and Nagoya and get to Kyoto for 14894 yen.
A JR-West (5-day) Kansai-Hiroshima Area Pass is, when booked in advance via the website, sold at 13500 yen. http://www.westjr.co.jp/global/en/ticket/pass/kansai_hiroshima/
So, (with your Hakone trip put aside,) your cost on this option is about 28400 yen, which is almost the same as the price 29110 yen of a 7-day (Ordinary-Class) Japan Rail Pass.
Merits of this option are
that you don't have to try to pack your major JR train rides within 7 consecutive days
and that you can take Shinkansen Nozomi and Mizuho trains between Shin-Osaka and Hiroshima, which a Japan Rail Pass does not allow you.
A possible demerit is that a JR-West rail pass is not valid on the Shinkansen section Kyoto -- Shin-Osaka, which is operated by JR Central, not JR West.
You can take a Special Rapid train (on the JR Kyoto Line) to go between Kyoto and Shin-Osaka or Osaka.
Shinkansen is available at Shin-Osaka Station, but not at Osaka Station.
According to the current Shinkansen timetables (, which will be revised in coming March),
when you travel with a Japan Rail Pass between Kyoto and Hiroshima, unless you catch a direct Hikari train in the early morning, you have to change trains. 09/07; Hiroshima / Miyajima
Regarding Miyajima Island:
On the 3 days from the 12th to the 14th of July 2018, the tidal range will be wide with both a high tide and a low tide expected to be during the daytime hours between 9 am and 5 pm.
Therefore these 3days would be better for you to watch the Grand Torii Gate of Itsukushima Shrine at different scenes, compared to the preceding 4 days (from the 8th to the 11th). Odawara to Hiroshima (Odawara Shinkensan Hikari) Hiroshima to Kyoto (Odawara Shinkensan Hikari)
Here "Odawara Shinkensan" seems to refer to Tokaido Shinka
(Shin = new; kansen = trunk line;
Tokaido (with "o" pronounced long) = traditional highway connecting Tokyo [Edo] and Kansai.)
There is also the Tokaido Line, which is a non-Shinkansen line; the JR Kyoto Line is a part of this line. to Itami Airport
Osaka Airport [ITM, Itami Airport] handles only domestic flights.
(Its official name still says "International" even though more than 10 years passed since its international services discontinued.)
When you leave Japan by airway from Osaka, your airport is Kansai Airport [KIX]. https://www.japan-guide.com/e/e2033.html?aFROM=2033_2158