Well the tree or stump must be dead, or you'd have growth showing. Even if you use a stumpkiller, that doesn't help it to degrade quicker - if you don't have the roots bored out, it takes some years for the stump and major roots to rot down.

sadly not a lot. enjoy the toadstools and imagine how all the nutrients from the rotting wood are feeding your grass etc.

They will in all probability keep coming until they have exhausted the food supply in the roots.They don't last long and lots of people enjoy seeing them (I do!) Treating what you can see makes no difference as the toadstools are just the fungal equivalent of flowers, and the main plant ( a mass of fine white threads) is buried under the lawn.