So you can't say "rest" without mentioning a tree? Finnish is funny in similar ways due to its ubiquitous forests. To "hunt" is "metsästää", "metsä" being "forest" the place you went to look for food.
You can say you metsästää for mushrooms (and other edibles) in the forest too, though, can't you? So maybe it's actually more of a direct correspondence to "forage". And, is that perhaps derived from / related to "forest"? (Or fodder?)
Not really, metsästää implies hunting for living things.
There's a dedicated verb for mushroom foraging, sienestää, or you can say sienimetsässä, "in the mushroom forest".