Hi!
Tokyo is your best bet when looking for anime and manga. Akihabara is nice, but I recommend "Mandarake" (you can google it, they have a site in English and maps). The one in Ikebukuro is especially for girls and nearby is also a Sanrio shop (Hello Kitty) and an Animate. Your daughter will love it!
Why don't you visit the Ghibli Museum?
There's also a Manga Museum in Kyoto!
If your daughter likes Osamu Tezuka, then there's a Memorial Museum not too far from Osaka, in Takarazuka.
For renting a car:
Yes, most signs are in English, but you need either a translation of your driver's license or an international one. It depends on your nationality, so please check yourself which one you'll need.
A JR ticket for 7 days? Are you maybe talking about the JR Rail Pass?
Also, I think doing Tokyo, Nara, Osaka and Kyoto in just 10 days might be a bit stressful.
Especially Kyoto will be packed with people during cherry blossom season, so you might not be able to move as fast as planned.