The forest could have markers indicating the track, could have bridges built to cross small streams, or even parts with paved (gravel or worse) paths.

Indeed, if the forest had not been tamed, not had some part of it claimed from the wild, it would be neigh-impossible to run in it.

That depends a bit on the kind of forest (some forests have way less ground cover than others) and on how you define “forest” (e.g. if a river that floods regularly flows through a forest, do you consider its shores to be in the forest?) and “running” (e.g. are the Barkley marathons (https://en.wikipedia.org/wiki/Barkley_Marathons) a running race?)

Good points, so the ice skating is a bit more wild with this perspective, if it's an "untamed" lake. (There certainly exists tamed lakes).