Here I will write the possible places we will visit in each city:

In general to find GF places in Japan: https://glutenfree-restaurant.com/

In general to find the train timetable: http://www.hyperdia.com/

Guide about the JR pass: http://www.japan-guide.com/e/e2361.html

Guid eabout 100 yen shops in Tokyo: http://www.jnto.go.jp/eng/arrange/travel/practical/100yen.html

Tokyo

Eating

Shopping

Kyoto

Guide to reach places in kyoto by train: http://jprail.com/destinations/kansai-area/guide-to-take-trains-in-kyoto-how-to-choose-the-best-route-by-train-to-get-the-major-spots.html

Eating

Osaka

How to move around Osaka: https://www.japan-rail-pass.com/japan-by-rail/travel-tips/osaka-loop-line

Osaka Castle and Park, Dotonbori, Amerika-mura, Shitennoji Temple.

Eating

Nara

Hiroshima

Eating