I went to Japan from late June to early July in 2016 and 2017 (and we did TDR in '17 for Tanabata Days.) In 2016, we didn't see Fuji-san at all. Every day we were someplace you could conceivably see Fuji-san it was clouded over. In '17 we got a lot luckier and saw Fuji-san on the way to Kyoto from the shinkansen and as a misty outline from Tokyo coming back from Koriyama. So you might see Fuji-san but anytime from June to September it's really not guaranteed and shouldn't be a deciding factor for a trip. (Now if your goal was to climb Fuji-san... July is a great time to do it. That's personally one of my goals.) It's amazing how well such a huge mountain can hide.
You can see a lot and do a lot in Japan in 10 days. The big decision would be is it worth it to get a rail pass especially since you want to do TDR. There are a lot of interesting temples and cultural sites you can reach with the much cheaper Tokyo Wide Pass, like Nikko and Kamakura. The hydrangeas are out then and Kamakura and Hakone have some spectacular ones. You can do a rail pass as well, but you really want to make sure to give Kyoto 4-5 days. It's worth it. We left for Kyoto on a 6:30 pm train after a full day in TDR. It's a matter of cost. Assuming both your children are under 12 but over 6, you're looking at 87,330 yen for 4 Japan rail passes, I think it would be closer to 30,000 yen for 4 Tokyo Wide Passes. There are additional passes if you want to overnight in Nikko or Hakone that might be worthwhile.
Most yukata you can buy for $100 are only going to be so well made. If you were interested in cheaper ones, the shrine flea market in Kitano Tenmangu in Kyoto always has LOTS of kimono and they can be as cheap as 1000 yen. They're all used, but that's sort of the fun of going. We rented high end yukata/kimono to walk around Kyoto in. It was a great experience because the place we used properly wrapped you up so you get the true experience. They had really nice kimono as well, though the children's selections were not as good.
In terms of English spoken, it depends on what your question is and where you are staying. For such an expensive hotel, I though the English language skills at the Tokyo Disneyland Hotel were disappointing. They were much lower than the skills at other similarly priced hotels. I felt like Osaka Universal overall had more English than TDR. TDR is awful for those with dietary restrictions. They've improved since '09, but are still very lacking. There are many places where there are English menus and English speakers in Japan, and there are of course places where lack of Japanese skills are more of a challenge. It depends on where you want to go. In general, the places most people recommend: Tokyo, Kyoto, Osaka, Hiroshima, Nara, Kamakura, Nikko, Hakone, Miyajima, Fukuoka, Nagasaki, Yokohama, and Kobe will be easier. Even places Takayama have a lot more English now, and the tour I took in 2010 in Japanese only is now offered in English as well. There are still places where there is less English but they tend to be places not on a shinkansen line or just not as popular with foreign tourists.
I've not been to Aulani, but I did stay at the Marriott K'Olina. The area is beautiful. We did the snorkel tour and my daughter is still talking about it. It sounds like you've all been to Hawaii before though and it's someplace you're more likely to be able to go back to in the near future.
You can see a lot and do a lot in Japan in 10 days. The big decision would be is it worth it to get a rail pass especially since you want to do TDR. There are a lot of interesting temples and cultural sites you can reach with the much cheaper Tokyo Wide Pass, like Nikko and Kamakura. The hydrangeas are out then and Kamakura and Hakone have some spectacular ones. You can do a rail pass as well, but you really want to make sure to give Kyoto 4-5 days. It's worth it. We left for Kyoto on a 6:30 pm train after a full day in TDR. It's a matter of cost. Assuming both your children are under 12 but over 6, you're looking at 87,330 yen for 4 Japan rail passes, I think it would be closer to 30,000 yen for 4 Tokyo Wide Passes. There are additional passes if you want to overnight in Nikko or Hakone that might be worthwhile.
Most yukata you can buy for $100 are only going to be so well made. If you were interested in cheaper ones, the shrine flea market in Kitano Tenmangu in Kyoto always has LOTS of kimono and they can be as cheap as 1000 yen. They're all used, but that's sort of the fun of going. We rented high end yukata/kimono to walk around Kyoto in. It was a great experience because the place we used properly wrapped you up so you get the true experience. They had really nice kimono as well, though the children's selections were not as good.
In terms of English spoken, it depends on what your question is and where you are staying. For such an expensive hotel, I though the English language skills at the Tokyo Disneyland Hotel were disappointing. They were much lower than the skills at other similarly priced hotels. I felt like Osaka Universal overall had more English than TDR. TDR is awful for those with dietary restrictions. They've improved since '09, but are still very lacking. There are many places where there are English menus and English speakers in Japan, and there are of course places where lack of Japanese skills are more of a challenge. It depends on where you want to go. In general, the places most people recommend: Tokyo, Kyoto, Osaka, Hiroshima, Nara, Kamakura, Nikko, Hakone, Miyajima, Fukuoka, Nagasaki, Yokohama, and Kobe will be easier. Even places Takayama have a lot more English now, and the tour I took in 2010 in Japanese only is now offered in English as well. There are still places where there is less English but they tend to be places not on a shinkansen line or just not as popular with foreign tourists.
I've not been to Aulani, but I did stay at the Marriott K'Olina. The area is beautiful. We did the snorkel tour and my daughter is still talking about it. It sounds like you've all been to Hawaii before though and it's someplace you're more likely to be able to go back to in the near future.