What should we do when people have a kid out of wedlock? Should the child just be a ward of the state?
The child becomes a ward of the state if both parents sign away their parental rights. They have no obligation to do so unless a social worker decides the parent(s) are unfit (as often happens, for example, if the baby is born with cocaine in her system). If they keep the child they can raise her in our outside of marriage. With one parent or two.
Should they be given the option of marrying?
They have that option.
If they decide not to marry, are the completely free of responsibilities to each other, the child, and society?
Nope. Unless they sign away their rights to the child, in which case the baby becomes a ward of the state, or (in the case of open adoption) (s)he may be adopted by the person/people whom the birth parents have chosen.
Are we not allowed to distinguish between this and adoption?
Between what and adoption? (See my answers above)
Are we not allowed to distinguish between issues that arise from biological inevitabilities and issues that arise from rare and occasional sexually perverse relationships?
The questions you posed above are all readily answerable. Who I have sex with and whether you approve of that person has nothing to do with whether my partner and I end up adopting a baby or conceiving one through artificial insemination.
How is someone's sexual perversion or related romances automatically the same thing as two people who have a direct part in having a child? We're not allowed to distinguish between these two things at all why?
I wholeheartedly disagree that the fact of my partner's gender makes our romantic relationship perverse. Marriage is about commitment, love, responsibility, mutuality, and family. This is true for gay and straight couples with or without children.
I freely admit that the difference between my relationship and a heterosexual relationship is that my partner is the same sex as I am. I also freely admit that this means we can't have children through sex with each other.
I don't see why our gender difference is a difference that should matter with respect to whether or not we can get married. You say that gender difference matters because if one of us was a man then we could reproduce and marriage is about regulating reproduction.
I've pointed out that that's not necessarily true, because (1) it is quite possible that if one of us was a man then we still wouldn't be able to reproduce because there's no guarantee that we are fertile; and (2) people reproduce all the time without being regulated by marriage and there is nothing in the law which requires them to get married before reproducing. In sum, because of (1) and (2), my argument is that the gender composition of my relationship -- while it might make you uncomfortable -- should not be a basis for denying us the ability to get married.