Can all animals of the same species crossbreed?
Species definitions are a somewhat contentious part of biology. There are no hard boundaries in nature that mean "this group here is one species, this group here is another species". Some people don't even believe that species truly exist and there are only gradations of relatedness. That being said, the Biological Species Concept is one of the more popular species definitions. 

From the linked website: "The biological species concept defines a species as members of populations that actually or potentially interbreed in nature". Thus, by definition, all animals of the same species can and do interbreed in nature. If two individuals breed in nature, they are considered the same species (under this definition). 

Of course, you can already see some problems with this definition. Under the BSC, are a dog and a wolf the same species? Do dogs and wolves breed "in nature"? Of course, fertile half-dog half-wolf animals exist, but does it even make sense to ask whether a domestic species breeds "in nature". I suppose you can say they can "potentially" breed in nature. But how do you define "potentially" breeding? Asking questions about species can lead to many exciting debates.