Some help with 3 week trip July'16

So this is how my schedule will look like atm, just a first version of it.

https://i.gyazo.com/32c2a4da2a5b065d9467f52d009955d3.png

A few questions

1. Will I need a 14 day JR Pass or 21 days?
2. How does my schedule look? Will I see a lot of Japan and are the stays in Hiroshima and Kyoto long enough to enjoy some daytrips to other places?
3. I would like to book some hostels, hotels and ryokans but it seems that most of the places are already fully booked.. Or is it not possible yet to book for next summer? Or are those Ryokans really that popular

1. 14 Days unless there are long trip during Tokyo period.

2. Depends how you define a lot of Japan, you might go back to Japan a dozen of times and still discovering new things or you might just rush through everything this trip and will not go back to Japan again because "it is boring".
Kyoto 4 days are pretty good. Hiroshima might be a little long unless you will travel out from Hiroshima for neighbouring district etc.

3. If it is July 2016, then you are WAY early. Most places will not have schedule up 6months before, some 2-3 months before. Even if some do have it, it is generic plan that has no discount or plan whatsoever (means you paying more).

21 Day pass would not pay off. The 14 Day pass, may or may not pay off. The thing about Japan is if you buy a long distance one way train ticket, every 100 km you travel it is allowed to get off the train line for a day. So say, if you buy a train ticket from Tokyo to Hiroshima, you can visit all the famous sites like Hakone, Osaka, Nagoya, Kobe, Okayama, Kyoto. You can't go to Kanazawa though as it is off the Tokyo-Hiroshima line. You would have to get a separate train ticket for that. So it is a bit of a grey area.

Your schedule looks good. I can do Hiroshima in a day but I am familiar with the area. Two days sounds good. Three only if you plan to day trip go to Okayama and Bunny Island.

Kyoto is an amazing city that takes a lot of time. Even more if you plan to day trip to Osaka, Uji, Nara, Ise. Osaka can be visited at night from Kyoto too. I would give Kyoto 3 days, Osaka 2 of those nights, and the last day as a day trip to Nara and/or Uji. Ise is a bit far and is mostly only if you are interested in how a Grand Shrine is like.

Next year's summer rooms aren't open yet. I recommend emailing or calling the hotel as a caution. Though, summer is not such a popular time to travel as autumn season is.

Try websites like Rakuten, Japanican, or jalan.net
They are Japanese sites but have English versions. Use these for hotels