If you can run to school and back I think that will be more productive training that brisk walking. The eccentric loading that occurs during running builds a very running specific type of strength that walking will not.
Concerning the sport specific training: You are correct that movement specificity is important. However even semi-sport specific movements that model most of the muscular recruitment patterns of your main sport will be very useful. The benefits gained from cycling or swimming will not transfer to running up hill. But hiking steep hills is not much different than running then. For your aerobic base training you will be wise to keep the intensity under AeT. That does not mean you can not do any uphill running. We suggest using short (10 second) sprints, done on very steep hills to build specific leg power. Later in the base period and beginning in the specific period you can add uphill running intensity workouts as well. It will be important to maintain a high volume low intensity uphill training though.