Thank you for this elaborated answer. No doubt Hokkaido must be amazing, with wild life and lots of space. But we are tipping towards staying closer to Tokyo and Kyoto because we only have 14 days and it's our first visit to Japan and it will probably take a full day to travel with train to Hokkaido. And another day to get back to Tokyo. So maybe Hokkaido will have to wait to next visit to Japan. The Kamikochi site may also be beautiful though it's more crowded? Thank you again!
There are numerous mountains and hiking destinations in Japan.
I have been to Shiretoko in Hokkaido and trekking there was just amazing.
You will get to walk the park which is registered as one of the World Heritage sites. Also, you will get to meet wild animals. I remember meeting so many Ezo (Hokkaido) deer! They were just walking around and not scared of people at all. If you go on a boat cruise, you will be able to encounter wild killer whales, too.
I could not find fulfilled information in English, but here is the page I found at japanguide.com : http://www.japan-guide.com/e/e6850.html
Rishiri Mountain is another famous hiking destination in Hokkaido. Rishiri Mountain is famous for alpine flowers, which Jun to August are the best season to see those flowers.
It may be far from your planning destinations, but Yakushima Island in Kyushu is also recommended personally. It is said to be the model place for "Princess Mononoke", a film by Ghibli Studio. The forests are amazingly beautiful, and I think this scenery is seen only in Japan.