First time Japan: 26 days, Tokyo > Alps > West-Japan

Hi, we are excited to visit Japan for the first time from september 15th to halfway October (exact date not sure yet). I am currently planning our trip and here's a first version. Could any of you please check out our itinerary and give feedback?

Day 1. home > Tokyo
Day 2. Tokyo
Day 3. Tokyo
Day 4. Tokyo (day trip to Nikko?)
Day 5. Tokyo
Day 6. Tokyo > Nagoya > Takayama
Day 7. Japanese Alps (best location for hiking?)
Day 8. Japanese Alps
Day 9. Takayama > Kanazawa
Day 10. Kanazawa
Day 11. Kanazawa - Kyoto
Day 12. Kyoto
Day 13. Kyoto
Day 14. Kyoto (day trip Arashiyama?)
Day 15. Kyoto > Osaka
Day 16. Osaka
Day 17. Osaka > Hiroshima
Day 18. Hiroshima (region)
Day 19. Hiroshima
Day 20. Hiroshima > Kumamoto
Day 21. Kumamoto (Mount Aso?)
Day 22. Kumamoto > Nagasaki
Day 23. Nagasaki
Day 24. Nagasaki > Fukuoka
Day 25. Fukuoka
Day 26. Fukuoka > home

(We might have an extra day or two.)

Can we do all of this by train? Should Kobe be in here? Are we trying to do too much at once? We're looking for a good combination of cities and nature/hiking. I understand it's too late in the year to climb Mt. Fuji, but what would be the best way to still experience this mountain in mid/late september?

Thank you.