It looks doable, though you are mostly just listing cities and not details.
You can certainly visit more than 2 castles - no two are identical. But you can easily skip Osaka castle - which isn't a castle, nor even a reconstruction - it's a museum with a castle facade over it. If you want to see a real one, go see Hikone Castle in Shiga - it is also one of the 12 surviving originals, like Matsumoto and Himeji. The Genkyuen Garden next to it is nice as well.https://goo.gl/FRe6t8
What you could do, unless you really want to spend a whole day in both Kobe and Himeji, is spend half a day to see Himeji Castle and the Kokoen Garden, and half in Kobe.https://goo.gl/65ZtRi
For Days 5-6, 2 days for Hiroshima & Miyajima should do nicely - you could likely also fit in Iwakuni.https://goo.gl/fAz4axhttps://youtu.be/0COOZXi66mM
For Hiroshima, don't miss the Shukkeien Garden and Mitakidera Temple - most do, which is a terrible shame. But at least the crowds are far less. Please note though that it will take you 3-3½ hours to get from Hiroshima to Nagasaki. You might want to break that up if you like - seeing the Yutoku Inari Shrine in Saga would be an excellent detour.https://goo.gl/dnqT25
Also you'll love Nagasaki - be sure to try the chanpon and saraudon for lunch dinner.https://goo.gl/hZVjBi
You might try to see Gunkanjima as soon as you can - it is a 3 hours trip, and if the seas are rough, the boat trips are cancelled.
Going from Beppu to Nagoya is really the sticking point - it is over 5 hours by train (although you could fly on ANA in just over an hour for only 10,800 on their Experience Japan Fare). You are also putting in over 4½ days in Tokyo, so you might consider adding another half or full day trip. For the Fuji area you need to choose either Hakone or the Fuji 5 Lakes, or add more time to see both.
One thing you might consider is if you can fit in a day from elsewhere, between Nagano and Kyoto, take a day and see Kanazawa. It's one of the nicest places and not overrun like Kyoto, with one of the nation's top notch traditional gardens.https://goo.gl/MkGZYp
If you have the chance to see some festivals, by all means go.https://www.japanvisitor.com/japanese-festivals/festival-october
Some areas like mountains might have some autumn colors, which are worth seeing.https://www.japan-guide.com/e/e2014_where.html