Mushrooms are the reproductive part of a fungus, a much bigger part of the organism known as Mcelium. fungus or Mycelium are the largest organisms on the planet. A mushroon is like the flower or fruiting body of the Mycelium and it's called a Sporocarp.
Sporocarps release spores which are like seeds. These spores will produce more sporocarps or mushrooms. If you want more mushrooms, tap vigously on the top of the mushroom cap to release the spores before you pick it. You can see the spores if you tap the cap on or over white paper. Mycolgist's refer to this as a sporeprint.
If you want less or no mushrooms you must provide the mycelium with an environment that it cannot tolerate. Most Mycelium are very sensitive to temperature, moisture and light. If you change one or more of the these you will terminate the mycelium. Copper is an effecive funicide that shuold be used with caution.
Understand that many animals can eat fiungus that will cause a horible death for humans. Not sure about dogs, I have never heard of a dog eating a mushroom, I suspect that a puppy would be much more likely in ingest a mushroom as they will sometimes eat things less apetizing, like dirty socks and dead insects. I have dogs and I have mushrooms and as far as I know the dogs have never eaten one. I did have a dog that decided to catch a toad once and he was halucinating and probably came closer to death than I realized at the time, as this particular species was a cane toad (bufo marinus). These toads are like many in that thier most effective defense mechanism is a toxic secretion.
When it comes to my pets, I am much more concerned with toxic amphibians than mushrooms.