I have a KML file of a number of polygon features. However, I would only like them to be visible above a certain distance from the ground when viewing in Google Earth, so effectively setting scale-dependent visibility. I would like to save this information to the file such that if I import the file into a different software, this information will be preserved.
In short, yes this can mostly be done using regions, as suggested in one of the comments.
Scale dependent rendering applies to 2D maps, where you're always looking straight down. In the 3D world of Google Earth, where you can tilt the view, there isn't a single concept of scale, since you can have things in the foreground at one "scale", with things in the background at a very different scale.
That's why Earth and KML use "regions" to specify when data is shown depending on location and angle of the view and data in 3D space. For example, if you have a set of tiles overlaid on the map, and you look at it from a tilted view, the tiles in the foreground can trigger a higher res version to load, while the tiles in the background can trigger a lower res version.
Exactly how you use regions will depend on what kind of data you want to show, and you're trying to do with it. But saving it in a file for transfer to other software probably won't work well, since most GIS software only deals with scale dependent rendering (one dimension instead of three).