Anarchists tend to direct their attention towards the role of the state as a primary perpetrator of oppression. However, some of the most oppressive forms of domination originate from civil society.
I think there's some semantics going on here that's debatable. Lenin describes "the state" as a body that exists to oppress one class in the service of another class, whether that be bourgeois against proletariat, or vice versa. Gramsci describes "the state" as a body that mediates conflict between the classes, but almost always favors one class over the other in those meditations, but a good mediator cannot function while totally favoring one side, hence why bourgeois state do often do small things that benefit the working class. Under both these descriptions, full communism would be stateless since class divisions would be abolished.
Now I suppose there is a debate to be had whether in this post-class society we'd still need some sort of administrative apparatus to mediate conflicts between individuals and to regulate anti-social behaviors, and whether that could be called "a state" depending on your definition.
Right, even if humanity manages to move past class society, there will still be need for administrative institutions. The need for organization, and conflict resolution exists independent of social classes. The key point Losurdo is making though is that is that such organizations aren't generally the key source of oppression and injustice in society.
Right, I think Gramsci's framing is the best, the state is a mediator between classes, but one that is HIGHLY biased towards the bourgeois. Most oppression happens via your boss or landlord, who usually have the backing of the state. However, there's plenty of instances of the state telling the bourgeois they've overplayed their hand and need to pay up. I've had two occasions of real estate court ruling in my favor against a shitty landlord, the government does sometimes punish corporations for dumping carcinogenics in the water supply. If the didn't the class contradictions would become too blatant and revolt would ensue.
Agreed, I think the idea of the state as a mediator is the most accurate formulation. It's also worth noting that the state doesn't just resolve conflict between classes, but within classes as well. For example, when capitalists can't agree on something they use the legal system to resolve differences.