Krellin,
You're right, I did make several statements without backing them up. Let me try to do so now:
To your first point: It's a bit difficult to argue that the State has an interest in stable relationships per se (though things like tax breaks suggest that such an interest may exist), but the State does have an interest in *recognizing* such relationships, for many of the same reasons that the State has an interest in recognizing familial relationships: property rights and inheritance, next-of-kin designations, legal visitation and spousal privilege rights, and so forth. These are important, not necessarily for the fiscal harm they impose or prevent directly, but for the ease-of-functioning of government. When a person dies, for example, without leaving a written will, it's important to know who is designated as next-of-kin -- that is, who is assumed to be meaningful in that person's life and the recipient of their possessions and outstanding obligations. A legally-recognized marriage is one way of designating that. Things like visitation rights for military families matter as well -- without being legally designated as related to a member of the military, you might not be allowed on a military base (for example). There are hundreds if not thousands of rights, privileges, and responsibilities that go along with being married, all of which are indications of the value the State places on recognition of a legally-recognized union.
As to stability itself, this is harder to argue, given that there are few (if any) legal provisions against rapidly marrying and divorcing. However, there is a reasonable implication one can draw from the types of benefits which are granted to legally-recognized unions: given that these benefits tend to revolve around property and familial connection, it can be assumed (but not proven) that such unions are intended to be relatively stable.
To your second point: Again, we must infer from the rights and responsibilities granted to legally-recognized unions. Such things as inheritance and spousal privilege against testifying, for example, quickly become unwieldy in unions of more than two people. Such a system could be open to far more abuse than our current marriage laws already are.
To your third point: The interests of the State regarding property and inheritance have no mention of gender for good reason -- gender is irrelevant. To the degree to which the State has an interest in the raising of children (which is a more tenuous case, but one which could be made by referencing, eg., Social Services and child protection/abuse laws), the evidence seems to indicate that children are best raised in loving, supportive, two-parent households -- but that the gender, sex, or sexual orientation of the individual parents seems to not have any meaningful impact. Thus there seems to be no reason for the State to have any interest in the gender or sex of two individuals in a legally-recognized union.
"How would it harm the state.... what difference does it make to the state if I am 'stable' or 'unstable'?" In your example, there really isn't an overriding state interest that should compel you to be in stable relationships, and there's no "harm" that I can see you doing to the state. I don't know where I implied that you were, but if I did, I revoke that implication.
"Any financial harm to the state (which is the only 'harm' you can do to the 'state') is because of the state's own invasive policies to begin with." I disagree here on two counts -- first, at least directly, you can harm the State in ways other than fiscal ones. Governments function to protect rights and provide services, and only one of those services is the regulation of financial interactions. Secondly, I would argue that there is not anything invasive about the rights and privileges granted through legally-recognized unions. It is just that -- legal recognition -- not anything invasive or prescribed. Individual couples come to the government for recognition, not the other way around.