Maybe, maybe not. You need an expert to ID them. You might try the folks at our local Oregon Mycological Society at http://www.wildmushrooms.org.

I doubt anyone can say whether or not a particular dog will eat them. My two sniff then leave.

Then too, poisonous to people isn't necessarily poisonous to dogs. And vice versa.

That said, you're likely to be more comfortable if you rake and discard.

Here is a link that might be useful: our Mycological Society

You might try aerating the lawn, it should improve drainage and at least reduce the number of mushrooms.

They resemble Coprinopsis atramentaria which I once cooked and ate -- tasted good, but there wasn't much of them. If you're eating wild mushrooms, someone once said that you ought to know the mushroom so well than you can call it by its scientific name. But if they're in my garden, I love the look and diversity of mushrooms that pop up. The only mushroom I might consider destroying would be a deadly amanita, and then only if I thought someone might eat it by accident, for the amanitas are beautiful.

Here is a link that might be useful: Coprinopsis atramentaria