trip to Japan summer 2015

my family is 4ppl and this is our first time in Japan and below is my plan for what and where to go and do when we there. Please kindly give us some advices and suggestion.
1. Travel in June or August which is better?
2. Which airport should we flight in/ out.
3. what transportation should we use for go place to place?
Please feel free to suggest any whether should or shouldn't do.
Thanks,
Tokyo Disneyland (3 Days) and place to see are:
Tokyo Tower/ Imperial Palace Plaza/ Tokyo Edo Museum/ Sensoji Temple/ Mt. Fuji
Kyoto/ Nara (3 Days) and place to see are: Ninja Museum/ Tenryuji Temple/ Bamboo Grove/ Togetsukyo Brige/ Golden Pavilion/ Todai-ji Temple/ Deer Park.
Osaka (3 Days) and pace to are:
Osaka Castle/ Shitennoji Temple/ Universal Studio

Travel in June or August which is better?

I am sorry to say that neither June nor August are very good months to visit Tokyo, Mount Fuji, Kyoto and Osaka. June comes with a lot of rain (rainy season) and August comes with very hot temperatures and large crowds (school summer holidays), especially in the theme parks such as Disneyland and USJ.

So, the question is what do you dislike most: rain, heat or crowds.

I personally would chose June and bring good rain protection. The theme parks are pleasantly uncrowded during June, and the temples of Kyoto can look very nice in rain.