Hi!
I'm just going to second what @Frog said about seeing Mt. Fuji from Hakone. I've been to Hakone 3 or 4 times and never seen Mt. Fuji from Hakone. And I am sure everyone is getting sick of me saying this, but I regularly see Mt. Fuji from Tokyo. I saw Mt. Fuji from the Disney Monorail on June 30, despite it being rare to see Mt. Fuji from Tokyo in the Summer. In November, seeing Mt. Fuji from Tokyo itself should be much more likely and is really quite spectacular.
It's important to note as well that it's ~35 minutes by shinkansen to Odawara. You then need to actually get to Hakone proper, and will depend on what you want to do there. It's all private lines and buses though, so if you're on a budget, think about whether you want to spend 2-3000 yen just on the chance you might see Mt. Fuji.
Be careful with the train to from Kanazawa to Takayama. It's about 2 hours but only if all your connections line up well. Also please note this page:
https://www.japan-guide.com/e/e5908.html where it states the Hidas from Toyama to Takayama will not be fully running until late November. It's possible they could finish track work earlier, but I wouldn't count on it. I don't remember where the bus schedule for the transfer is now, but if you take a local part way and then transfer to bus I know from the Takayama direction there was a 20+ minute layover. I believe there are direct buses from Toyama to Takayama that are covered by the JR Pass during this hiatus, but I don't know the times. But I would spend some time researching this, because in general, trains from Toyama to Takayama are not as frequent as from Nagoya to Takayama. if you are visiting Shirakawa Go, you might want to just continue on by bus to Takayama.
So if it were me, I would do:
Day 1 (Tue): Arriving in the morning at Narita. Staying in Tokyo. (night Tokyo)
Day 2 (Wed): Tokyo (night Tokyo)
Day 3 (Thu): Tokyo (night Tokyo)
Day 4 (Fri): Tokyo (night Tokyo)
Day 5 (Sat): Tokyo (night Tokyo)
Day 6 (Sun): Morning Tokyo --> Kanazawa (~3h train) (night Kanazawa)
Day 7 (Mon): Kanazawa (night Kanazawa)
Day 8 (Tue): Kanazawa --> Visit to Shirakawago continue to Takayama (night Takayama)
Day 9 (Wed): Takayama (night Takayama)
Day 10 (Thu): Morning Takayama --> Kyoto (night Kyoto)
Day 11 (Fri): Kyoto (night Kyoto)
Day 12 (Sat): Kyoto --> Visit to Nara (night Kyoto)
Day 13 (Sun): Kyoto (night Kyoto)
Day 14 (Mon): Kyoto --> Hiroshima (night Hiroshima)
Day 15 (Tue): Hiroshima --> Miyajami --> (night Hiroshima)
Day 16 (Wed): Hiroshima --> Himeji castle ---> (night Tokyo)
Day 17 (Thu): Tokyo (night Tokyo)
Day 18 (Fri): Return flight.
This would give you 5 nights in Tokyo, 2 nights in Kanazawa, 2 nights in Takayama, 4 nights in Kyoto, 2 nights in Hiroshima and 2 nights in Tokyo at the end. If you wanted you could maybe take 1 night from Tokyo at the beginning and give it to Kanazawa or try 4 Tokyo, 2 Kanazawa, 2 Takayama, 4 Kyoto, 2 Hiroshima, and 3 in Tokyo. If you really have your heart set on Hakone if you're not planning to stay in a ryokan you could even go there on you way from Himeji to Tokyo on day 16. There are direct trains to Narita from Ofuna which is really not that far from Hakone. Or do the 4 (Tokyo) 2 (Kanazawa) 2 (Takayama) 4 (Kyoto) 2 (Hiroshima) 1 (Hakone) 2 (Tokyo) Otherwise you are arriving in Kanazawa on a Monday and some of the museums will be closed. We did Kanazawa on a Monday, but I also knew my daughter didn't necessarily want to go to the art museum. If you change to the 4-2-2-4-2-1-2 itinerary or 4-2-2-4-2-3 you would have a Monday in Shirakawa Go which is unaffected and on the next Monday you would be in Miyajima and only the History Museum would be closed.
AirBnB since this Summer has had a lot of problems in Japan and I've heard people are making up fake certifications. In June they were compensating people who had their stays canceled on short notice, but I do not know if they are still doing so. If Kyoto is too expensive, I would probably stay in Osaka or Kobe, since you're going to want to head that direction to Hiroshima vs. Nagoya which is closer to Tokyo. Though it feels like there are constantly new hotels opening up in Kyoto. I've gone there 3 years in a row, and each time I've booked there have been new hotels. I don't know your budget for hotels so can't give any advice.
I hope this helps!