The human body would adapt and deal with elevation. Many top athletes often train at high altitudes to help train their bodies to absorb oxygen more efficiently from the thinner air.
If the terrain was pure mountains then the ability to adapt to the local geography would be key. Natural shelters in the forms of caves for early survival, though a medieval culture would be able to extend and possibly even create man-made caves for habitation over time. This will help them deal with the cold and natural predators as our, and their ancestors did. Fires at the entrance to the cave would ward off predators and help keep the cold at bay.
The most significant hazards would be natural. Snow and ice would create dangerous living and working conditions. The constant need for food would also be a source of difficulty, with meat being a large part of the diet. For the survival of a large settlement, farming is necessary, though in mountains this will be near impossible limiting settlements to a small size.
Cave ins, earth quakes, avalanches, all sorts of natural disasters would affect their daily lives. We struggle to deal with these in this day and age. They would simply be a fact of life for those in the medieval age that they had to contend with and get through. Any one serious act could end the settlement though.