iirc some mushrooms do that without the corpse - the main mushroom is in the center underground, while the fairy ring is basically the fruit it produces (these are probably not the correct words to describe this - but thats the principle, I think :D)
The mycelium is the main body of the fungus, yes, and it's the entirety of the circle. The mushrooms on the edges are indeed the fruit.
And it's every type of mushroom that does this... sort of. Any mushroom that grows along the ground at least. There are plenty of varieties that grow on trees and throughout a leaf pile or such.