I'd say you have a pretty well balanced itinerary over all, but a few impressions...
For Kamikochi, I think you are in the ball park for opening day, but AFAIK the official first day has not been announced yet. I suggest you keep your schedule flexible.
For Tokyo, unless you really hate big cities, I'd say definitely put in 3 days for it. And with that, you are already seeing Tokyo, you can largely skip Osaka during the day. After the temples of Kyoto close down around 5PM, you can zip over to Osaka for the evening. At night the city comes alive and has some great places to see, such as Dotonbori
https://www.youtube.com/watch?v=COJ_MeXbeowplus the night views from the Umeda Sky Bldg
https://www.youtube.com/watch?v=xb1wuyAGgyQand Abeno Harukas Bldg
https://www.youtube.com/watch?v=-AyUZmGbW8g are wonderful.
Instead, take some day time and go see Kobe - it has a nice atmosphere and several great places to visit.
For your Day 9 there is really not enough time to see both Himeji and Hiroshima in a day. Stick with Himeji and see the castle, Kokoen Garden next to it, and if you can, go up to see Mt. Shosha.
For Day 10 you can see the main sights of Hiroshima and Miyajima is you have an early start. Everyone goes to see the Peace Park/Museum, but don't miss the Shukkeien Garden and castle. For Miyajima, the best sights to see are the Itsukushima Shrine, Daishoin Temple, and Mt. Misen.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-iCNZ6AdsKGUB-HPKNxP1...If you had a little more time, Iwakuni is a stone's throw past Miyajima and has a unique bridge and mountaintop castle.
https://www.youtube.com/watch?v=0COOZXi66mMFor Day 13, you might find Takayama rather touristy - but don't miss Shirakawago. It only takes a few hours to see.
For your time in the Tokyo area, if not too tiring you might group your day trips together and use a rail pass or regional pass for them. If you see Hakone there is the Hakone Free Pass. For Tokyo if you spend 3 consecutive days there, a 3 day subway pass is easy to make worthwhile.
http://www.tokyometro.jp/en/ticket/value/travel/index.html#anc03