Marriage need not be a "government thing" in the modern sense, with all the legal distinctions we currently have. Marriage is inherently a legal thing, in the sense that it's a contract between people.
Marriage confers very special rights, such as permanent residency or social security inheritance, that you cannot obtain through private contracts. Therefore the government has to be involved.
this is the crux of the issue in that government policy is, in large part, based upon a private contract. neither government nor judeo-christian sects (which, if we're honest, is whom we have in view when we speak of "religion") seem to be able to grasp the full implications of this.
It depends on your definitions. If you consider all legal systems to be "states," including primitive tribal ones, decentralized ones, etc., then yes, "state is a prerequisite for law" is a tautology. But today, "state" usually refers to a relatively large and relatively centralized organization governing a well-specified region that is recognized by other such states. Using that definition, a state is very clearly not a prerequisite for law.
The context that allows a system of laws to operate doesn't just spring forth spontaneously from a vacuum. Rather, it exists because we have a government based on the rule of law.