Hi Helen!
Hiking in Japan is one of my favourite topics :).
Firstly, you should take care as you're planning to travel right at the end of the hiking season. Kamikochi closes in mid-November for the winter, and certainly other locations will be getting colder, with changeable conditions being the norm, there's even the possibility for backcountry gear to be needed from mid-Dec onwards, so 'be prepared'.
That said, if you're a seasoned hiker then you should be fine and will be rewarded with vivid autumn colours until the middle of November. I recommend you purchase the Lonely Planet book: 'Hiking in Japan' for a more detailed resource - I've found it invaluable in the past.
For Koyasan and Kumano, have you considered actually doing one of the [World Heritage pilgrims walk between Koyasan and Kumano called Kohechi]? It takes around 4 days, and so would possibly fit into your plans. [Here is a hiking report] from someone who walked it towards the end of October 2009.
I'll leave you to mull that over, and assume that you'll need public transport between each of the locations you list to get an idea of total rail travel prices. Please note that I have not included non-JR legs, such as the trip from Osaka to Koyasan, and I've done my best to put you closest to the areas you list, but the final stations you use may differ:
- Tokyo > Matsumoto: ¥6710, (170mins, 235km)
- Matsumoto > Iseshi: ¥8230, (241mins, 296km)
- Iseshi > Shingu: ¥5320, (187mins, 152km)
- Shingu > Osaka: ¥11430, (291mins, 421km)
- Osaka > Kyoto: ¥540, (29mins, 42km)
- Kyoto > Yonago: ¥11150, (217mins, 378km)
- Yonago > Hiroshima: ¥9160, (190mins, 320km)
- Hiroshima > Aso(kumamoto): ¥14620, (187mins, 449km)
- Aso(kumamoto) > Kagoshima: ¥8910, (152mins, 223km)
- Kagoshima > Uno: ¥21450, (313mins, 766km)
- Uno > Tokyo: ¥17180, (305mins, 765km)
TOTAL: ¥114700 for 11 legs
As you can see, with a 21 day pass (¥57700) you'll save a whopping ¥57,000 over purchasing individual tickets. I see based on your nights that you're planning on staying for 36 nights (maybe I'm reading it wrong?). Of course for a 21 day rail pass to work, you'll have to fit your travels into 21 days, so this may also influence your choices about what or what not to include.
Even if you were to travel for all 36 nights, based on your projected savings, you could purchase an extra 14 day pass (¥45,100) and still make minimum savings of ¥11,900. (if you chose this options you could also include a trip to or from the airport and other JR travel in addition to your itinerary to make further savings).
For Aso, make sure you read this post on the best way to visit. I have a feeling you'll like the Higashi platform and hike round to the Nishi platform the most.
Spacing wise, I think you maybe are fitting too much in, and possibly I'd look at missing Mt Daisen (which is lovely, but perhaps a little far out of your way all the way over on the sea of Japan coast), and replacing with a trip down to the hotsand baths Ibusuki and nearby kaimon dake mountain after or before Yakushima.
Hope this helps, and feel free to ask more questions on this. As I say, it's my favourite subject :)