A clothing rack is the place that hangers are able to hang from. Hangers themselves are the triangle shaped item for shirts and coats for example. Also a rack can be for other things, like shoes, cookware, or dishes. Hangers are primarily for clothing in that context. (An exception would be 'airplane hangers' which are the buildings that airplanes are parked in.) A 'rack' could also be referring to a place to hang any other items from a hook. Track, footpath, and path are less distinct. Footpath is more like a place people normally go walking, on foot. Path is also marked in some way, to be distinguished from the areas not walked on. A track would refer more to something fixed (train track, running track for exercise). To make a new way to walk through deep snow would be the phrase 'cutting a path through snow'. If it was an animal or just walking on fresh snow, it would be more like 'making tracks in the snow'. To go into the forest and pick mushrooms can be called 'going mushrooming'. I would caution you though that many American English speakers might be confused since it is not common to do this. If one were to chop down a tree, they would cut it into logs with an ax or saws. Wood is when we speak of the material itself, examples: a wooden table, purchase wood planks, wood floors. The term wood is for the material that has been cut and is useable for a purpose. Firewood is the logs cut with intention to burn them. Specific to using them for fires instead of for building things.

Обычно в таких случаях я использую https://dictionary.cambridge.org Там хорошее описание и даётся несколько примеров, где слова используются в контексте. Так же есть ещё один сайт, где вы можете посмотреть слова в контексте http://context.reverso.net/перевод/