I don't know about the US varieties, but the spores are so small and light, and produced in such huge quantities, I think most are universal. I have never heard of an animal poisoning themselves. Over here the ones that grow on grass are not poisonous, though there can be buried wood. Pets all originate from wild animals, and, be they anything from dogs to tortoises, I can't imagine them eating anything naturally occurring that would be harmful to them, that's a human sort of trait.
The spores produce a mycelium, fine, white strands, which have single sets of DNA. When two mycelia grow alongside each other they can join and produce a new one with two sets of DNA in it, this is what pushes up the fruiting bodies we see as mushrooms or toadstools, not those directly originating from the spores.