Hakone Pass and Japan tour plan suggestions

Hello everyone,
In August I am visiting Japan. I have made my very own plan for this visit as you see below :)
Generally I wanted to ask about one thing specifically: Does Fuji Hakone Pass works the same as Hakone Free Pass in the Hakone Area? I plan to do Hakone Round Course, and I am not sure if it covers Moto-Hakone to Hakone-Yumoto bus ride, as well as Gora Park, do anyone of you know this? I prefer Fuji Hakone Pass because I am visiting Fuji 5 Lakes the day after Hakone, so it seems it will be valid for both of those places.
Regarding the plan - can you take a look and see if the places are in your opinion worth visiting? Also, am I not rushing too much? :) I have even put the hours I plan to stay at one place before I move to another. The time between each attraction is time given by Google Maps to move from one to another spot. I will probably get lost few times, so I think I should add some time to this. If you have any suggestions I am open for modifications :) I will sleep in Kyoto and Tokyo only.

PART I. Kyoto (Kyoto Sightseeing Card)
8.08
Nijo Castle (Nijojo) 8:00 - 9:00
Sento Imperial Palace 9:20 - 10:30
Kyoto Imperial Palace 10:30 - 11:30
Ginkakuji (Silver Pavilion) 12:30 - 13:30
Kinkakuji (Golden Pavilion) 14:30 - 15:30
9.08
Fushimi Inari Shrine 7:45 - 9:45
Katsura Imperial Villa 11:00 - 12:00
Arashiyama and Sagano
▪ Sagano Scenic Railway 13:00 - 13:30
▪ Hozugawa River Cruise 14:00 - 16:00
▪ Monkey Park Iwatayama* 16:10 - 16:45
▪ Bamboo Groves 17:00 - 17:30
10.08
Higashiyama District
▪ Kiyomizudera Temple 8:35 - 10:00
▪ Yasaka Pagoda 10:15 - 10:35
▪ Kodaiji Temple 10:45 - 11:30
▪ Maruyama Park 11:40 - 12:20
▪ Yasaka Shrine 12:20 - 13:30
▪ Chionin Temple Sanmon Gate 13:40 - 13:50
Nanzenji Temple 14:10 - 15:10
Heian Shrine 15:30 - 16:30
11.08
Himeji
▪ Himeji Castle + Kokoen Garden 9:00 - 12:30
▪ Steak Senju 13:00 - 13:45
Kobe
▪ Hakutsuru Sake Brewery Museum 15:15 - 16:00
12.08
Mount Koya (Koyasan World Heritage Ticket - Shin-Imamiya Station)
▪ Okunoin Temple 10:30-13:30
▪ Garan 14:00 - 15:30
13.08
Nara
▪ Horyuji Temple 8:10 - 9:10
▪ Todaiji Temple 9:45 - 10:45
▪ Nara Park 11:00 - 13:00
▪ Kasuga Taisha 13:15 - 14:15
14.08
Kyoto National Museum 10:15 - 11:15
Rengeoin Sanjusangendo 11:20 - 11:50

PART II. Tokyo. (Tokyo Subway 48-hour Ticket)
15.08
Tokyo Imperial Palace 10:00 - 11:15
Tokyo National Museum (Ueno Park) 12:00 - 14:00
Tokyo Skytree 14:25 - 15:00
Shinjuku
▪ Kabukicho 15:35 - 17:00
16.08
Asakusa
▪ Kaminarimon (Kaminari Gate) 9:00 - 9:20
▪ Kimono Rental 9:30 - 9:50
▪ Sensoji Temple 9:35 - 10:15
▪ Asakusa Shrine 10:20 - 10:50
▪ Nakamise Shopping Street 10:55 - 11:35
Meiji Shrine + Inner Garden 12:25 - 14:00
Shibuya
▪ Center Gai 14:15 - 15:15
▪ Hachiko Statue 15:20 - 15:30
17.08
Nikko
▪ Toshogu Shrine 9:30 - 10:30
▪ Iemitsu Mausoleum (Taiyuinbyo) 10:45 - 11:45
▪ Tamozawa Imperial Villa 12:00 - 13:00
▪ Kanmangafuchi Abyss 13:15 - 14:00
18.08
Hakone (Fuji Hakone Pass - Odawara Station)
Hakone Round Course
▪ Gora Park 9:35 - 10:15
▪ Owakudani 10:45 - 11:25
▪ Lake Ashinoko (Ship cruise) 11:45 - 12:15
▪ Hakone Shrine 12:45 - 13:15
19.08
Fuji Five Lakes (Kawaguchiko - Fuji Hakone Pass)
▪ Chureito Pagoda 9:00 - 9:30
▪ Bat Cave 12:00 - 12:30
▪ Iyashi no Sato 10:30 - 11:30
▪ Wind Cave 12:00 - 12:45

The 09.08 day is too much. You haven't allowed enough transit time between major locations. And the Monkey park will take linger than 35 minutes. It takes that long to walk to the top, and the area where the monkey are needs at least 30 minutes.

Yeah, this particular day is tough indeed :) So it might be wise to move Fushimi Inari after the Bamboo Groove, since Fushimi Inari is open 24/7?

Hello again,
I have another question about Fuji Hakone Pass, on the japan-guide site it is shown the pass covers buses aaround Lake Saiko (https://www.japan-guide.com/e/e2358_014.html) and on Odakyu site the map shows that the pass does not cover Lake Saiko buses (https://www.odakyu.jp/english/passes/fujihakone/). Can anyon confirm which is true?

The main website of any business is the one to rely on. Not a third party website.

First of all, yes, Fuji Hakone Pass works exactly the same as Hakone Free Pass in Hakone area. And yes, it covers the bus rides as you mentioned, but just make sure you take either Hakone Tozan Bus or Odakyu Bus, not Izu-Hakone Bus (this one is not covered by the Fuji Hakone Pass).

As for the Fuji Part, yes it covers Lake Saiko area too. In fact, the Green Line of Omnibus circles around Lake Saiko area. The omnibuses are covered by Fuji Hakone Pass.