Honestly, I would recommend doing a bit of reading about AirBnB in Japan from Summer 2018. Japan changed its regulations and some people have been cheating and used fake certificates to get back on the site. I suspect the prefectures/cities that don't want AirBnBs might also crack down, so just be aware.
As for your itinerary...
I think if you want to go to Fuji Five Lakes, you're better off taking a direct bus from Tokyo rather than taking a train 1+ hour to Mishima and then transferring to a bus for 1.5 hours. The buses out of Tokyo run 2 buses every hour from Shinjuku and ~2 buses every hour from Tokyo Station. I believe the bus from Mishima to Fuji Five Lakes is a lot less frequent as are Hikaris that stop at Mishima and Kodama shinkansen will add onto your travel time. Coming from the US, you will have the advantage of jetlag working for you, and can get out to Fuji Five Lakes bright and early, but you are also going to get tired in the evening and you're looking at: 2 hours to Fuji Five Lakes, exploring Fuji Five Lakes, and then another 3.5 hours onto Kyoto. I think you are also underestimating crowds for the Gion Matsuri. You can't get there at starting time and hope to have a decent spot. I guess depending on when your flight arrives and if you think you and your group could handle it, you could technically go direct from the airport to Fuji Five Lakes. That would also increase your chance of seeing Mt. Fuji since you would be there in the morning when the sun rises. It can be hard traveling onwards when you first arrive in Japan, but based on everyone's ages it could work. I use to do it as a solo traveler from the USA. It did not work well when I tried it last trip, but my child was 7 at the time.
You really only have the days of the 16th and 17th in Kyoto. I really like Nara, but I'm not sure you really have the time to visit Nara as it will take up a whole day just giving you 1 day for Kyoto.
I personally wouldn't change hotels for your Osaka day. It's easy enough to go to Osaka as a day trip and I honestly wouldn't want to have to find a locker ever again in Osaka or waste the time going to a hotel or lodging first. We spent a good 40 minutes trying to find a locker in Osaka Station last trip. We eventually had to use Sagawa, which for our purposes was cheaper and easier but even so... I don't think it's worth it in your case. It's easy enough to find a Hikari from Kyoto that goes to Himeji or Okayama. I noticed most of the Hikari I took ended in Okayama.
Getting tickets to the Ghibli Museum has definitely gotten harder it also has gotten a lot more crowded. We went in July 2018. We couldn't get tickets through JTB, but our hotel kindly got tickets for us.
I've only been to Hokkaido once and only visited Hakodate and Onuma so can't speak for your Hokkaido plans.
I believe Kyoto has some Machiya hotels now where you can rent a machiya townhouse that is grouped with others to make a little lodging area. I saw some for rent last trip, but as we are a party of 2 it seemed overkill. I've also seen a few interesting looking apartment hotels like Hotel Gentle Fox and a place out near Nijo Castle whose name slips my mind.
Here are the bus times from Tokyo Station to Fuji Five Lakes:http://bus-en.fujikyu.co.jp/highway/detail/id/37/
Here is Shinjuku to Fuji Five Lakes:http://bus-en.fujikyu.co.jp/highway/detail/id/1/
vs. the buses to and from Mishima:http://bus-en.fujikyu.co.jp/highway/detail/id/29/