The phrase you cite is "endlessly amusing" because it is a contradiction in terms; communism/socialism is predicated on the presence of a government, and anarchism is predicated on the lack of one. If you need that explained, I'm not sure why you're participating in this thread.
It's really not. Socialism is about collective ownership of property, state socialism is only a subset of socialism. But perhaps you could explain why prominent anarchists like Bakunin aligned themselves with socialists when their ideals were so diametrically opposed? It seems like you're using rather ahistorical definitions of anarchism and socialism.
The distinguishment between that hypothetical entity and the 'state' as we know it is certainly up to debate, but I wouldn't therefore say that anarchism and communism or socialism are therefore contradictory.
How can we debate the necessity of the existence of government when we can't even decide what a government is?
A government can only exist when it has a monopoly on the legitimate use of force in most situations. How can that, where only one group can use violence without being punished for it, possibly fail to impose a hierarchy?
We can have a government because we can set expectations and behavior. Only when we have police and a state apparatus do we then become a society whose mode of governance is via the state.
I'd say that 'government' would be nothing more than a debating society, like the League of Nations, and just about as useful in preventing, say, homicide as the League of Nations was at preventing WWII.