Research a holiday in Asia

Me (30yrs)and my girlfriend(27) are going to Asia in July. We want to visit Hong Kong, Beijing or Shanghai( we liked to see the Chinese wall), Japan (Mostly Tokyo and hike Mt Fuji) and maybe Seoul instead of Hong Kong if its fits better. We like hiking, shopping, good restaurants and skybars. Not so into museums or art. Our holiday can start on Friday 29 (Or 30 Saturday)June from Oslo, Norway and we need to back a latest 20 July but if the flights are better earlier thats ok. (Back to Oslo). We want you to find good flights with few connections and a total price under 1300$ ish. To find good accommodation in every city on avg. 100$ per night. We also want to go on hikes in China and Japan, as mentioned we wanna see the Chinese wall and hike Mt fuji. Maybe there is something more we are missing? We are very flexible and wanna se as much as possible, a day tour of every city would be nice but one day is usually enough. I heard that the train ride from Tokyo to Kyoto is something we shouldn't miss. Is that a possibility?