It depends who you are using to book your airfare. I frequently use JTB (Japanican I believe is a subsidiary of JTB) to purchase my airfares and I usually do it 2-3 months out. In general, airfare is better if you are near a major city with direct flights to Japan. (So the best fares are from LAX and SFO, and you can find all right fares from NYC, Chicago, Seattle, and a few other cities.) I've flown direct from NYC and done a few plane changes in SFO, Chicago, and LAX when going to and from Japan. Usually I had changes because I wanted a specific airline or I was looking to fly into or out of an airport besides Narita and I don't live near SFO or LAX.
In your case, if you are continuing onto Hong Kong or coming from Hong Kong you might want to look into flying into/out of an airport besides Narita or Haneda. So for example, fly into say Fukuoka from Hong Kong and then work your way East to Tokyo and Tokyo Disney and then leave from Haneda or Narita for the USA. Or fly into Narita/Haneda from the USA and then fly out of Kansai Airport to Hong Kong. Generally, it doesn't cost that much more to do an open jaw flight and you will save money in transport costs within Japan unless you plan to just stay in Tokyo.
I know this is not a popular feeling on this board, but I couldn't take 4 days in TDR. The parks are nice and well done, but there is just so much to see and do in Japan I would be miserable giving up 4 days of my trip to Japan to be in TDR when there is so much more I could be doing elsewhere. I definitely like giving a day to each park, but more than that I just can't do.
My DD loved TDR. She's currently 6, turning 7 soon. TDR is very different from WDW and DL, which she has also been to. There are things I love about TDR but there are other aspects I find disappointing. We're going to go back in late June or July of 2018 and I am debating staying at the Tokyo Disneyland Hotel again or just staying in Tokyo because while I loved the 15 minute early entry, I was so disappointed by the actual hotel. The beds were awful and I thought the bathroom looked dated.
I am a regular contributer to Japan-Guide. Their forum has some really great information but I'd recommend spending time on the website first and doing some research. There are a few members there who've gotten a little snarky with people who haven't done any research first.
For booking hotels/ryokan I use: the actual hotel's website, Booking.com (because they include all taxes), Japanican (they have some better rates for luxury properties), Rakuten and I've used Jalan a few times in the past. A lot more places have English websites than once did.
The Ghibli Museum is very popular. You need to try to book ahead of time with JTB and pay their premium. If you can't get your date that way Japanican offers a "tour" with an even higher premium for the tickets. I wet a long time ago, it was fun. I plan to take my daughter next trip.
I think if you're going to be in Japan for 2 weeks you really should try to make it to Kyoto and Nara. Kyoto and Nara are also good places to stay in a ryokan. I like Hakone, and if you are there in late June, you are there for their big hydrangea time (and the special hydrangea night trains) but I am not sure how interesting the area is for kids. When we were there in June 2016 it poured and Owakudani was closed for walking around because of volcanic fears. I also just have not yet found a hotel I like there. I've stayed at several.
I really Kamakura and I enjoyed Nikko. But again, if you have 2 weeks there are a lot of other places you can also visit. One of the people on Japan-Guide whose view I always appreciate loves Enoshima as a place to stay.
I think the big challenge for you, might be accommodations. I am not sure how many of you will be going, but just so you know most accommodations will count anyone over 12 as an "adult." There are AirBnBs but be warned, in many municipalities they are not legal and your reservation can be cancelled at the last minute.
As you just say Summer, I would avoid going in August, because it's Obon and things in general are more crowded and booking trains can be harder because it is a holiday time in Japan. It's a little hot and it can be rainy, but I've enjoyed the 2 trips I've now done late June to early July. My daughter will be doing trip #3 in 2018. We're still working on destinations because I can't decide if Naoshima is worth it or if we are better off visiting Ritsuin and the caves at Akiyoshidai.