Hi!
Sorry, my child has been off school and we went on a vacation, so I've not been responding. I can't speak for other posters.
First, I don't know where you are from, but many people find the heat and humidity in Japan in mid to late June, July and August incredibly oppressive. I've been going Summer the last 4 years and it really can vary from being OK to just miserable and I have a relatively good tolerance for heat and humidity unlike many other forum posters.
Unless you've pre-booked to stay on property I would switch your TDR days to a two day period between July 7-9. This is the generally recommended crowd calendar :
http://www15.plala.or.jp/gcap/disney/ You are much better off going to the lightest colored days as possible for both parks. Sea in particular gets insanely crowded though Land will have the new B+B ride.
Please be aware, unless it is bad weather, even on not crowded days the parks, especially Sea, get super crowded and make sure to give yourself time to get there. It's commutable from Tokyo, but it takes longer than you would think. If the park opens at 9 people will be there in line at 7:30 am or earlier. If you have jet lag and are up early take advantage of that. I like all the Tanabata Days merchandise.
Generally, with USJ they work out their themes ahead of time and I am not sure they are going to switch things up in early July. The crowds can be awful there and they have a crowd calendar as well.
http://www15.plala.or.jp/gcap/usj/ Though it can take a while for it to update for your actual dates vs. TDR is updated for June and July already. Express passes can be wonderful. We used them one trip. The other trip it is pouring rain so a non issue.
I think it would be possible to get from Miyajima to Naoshima depending on when you leave and what you want to do on the way and do in Naoshima. The same is true for Naoshima to Kyoto. Just have low expectations. Last I checked, there were no children's bikes available on Naoshima so you will be relying on the buses or walking and it will be hot and humid.
I think Nara and Fushimi Inari in one day is a long day and I am generally a faster paced tourist. I think it can be particular hard with heat, crowds, and traveling with children. I can honestly say, that unless your children are seasoned travelers who thrive in heat and humidity, I think you need to find one day with less activity. I know for mine, we took one day to relax and do karaoke and crafts at the Kyoto Handicraft Center.
I can not imagine anyone spending a full day in the Ghibli Museum. It is tiny. Make sure you know how to get tickets and get yours in advance. There are different possibilities depending on your home country. My kid loves Miyazaki films was was not wowed by the museum. I think it is most popular with people who want to buy lots of merchandise from the museum gift shop.
I can't answer for your Olympics questions as I've never seen them and have zilch desire to see them. I want to be out of Japan waaaaaay before they start.
Good luck!
BTW, do not assume you will be able to use lockers in the train stations. I don't know if they will do the same thing for the Olympics, but for the G20 this past Summer, all lockers and trash bins were closed for several days surrounding the G20 through out Japan. I was in Tohoku during it and lockers were still closed there even though it was nowhere near where the Summit was in Osaka.