Karl Marx said that capitalism commodifies everything. So I'm wondering if that rule applies to the very State itself.
Yes. For example, the nationalisation of industries brought expanded reproduction into the state function. So do public private partnerships. The generalisation of private sector management standards and language have made obvious the service provisions of commodities to customers in areas like welfare, defence procurement, archival or parliamentary services.
I commend Volume 1 of Capital to you, in particular chapters 1 to 6, because your use of the term in commentary differs significantly from that present in the literature you reference. The value-form, as in "the commodity," is a social relationship not an object.