Do I need JR Pass?

Hello, my boyfriend and I are traveling to Japan in September this year and I would be grateful if someone could tell me how I can check if buying a JR pass for 7 days makes sense for the following itinerary:

Day 1. Tokyo - Matsumoto
Day 2. Matsumoto - Shinanoomachi-Tateyama-Toyama
Day 3. Toyama-Kyoto
Day 4. Kyoto - Arashiyama - Kyoto
Day 5. Kyoto - Himeji - Kyoto
Day 6. Kyoto - Nara - Kyoto
Day 7. Kyoto - Hiroshima
Day 8. Hiroshima - Miyajima
Day 9. Miyajima - Tokyo

From what I found JR pass is valid on most of the trains here except to/from Miyajima - is it correct?
If it is correct then probably it makes sense to activate the pass on day 1 rather than 3?
Also, how can I check how prices for separate tickets in my itinerary add up in order to see how much I save by using JR pass?

Thank you!

Galina