1) Even without objective morality, human societies will form "social contracts" and behave, well, just as they actually do.
True. Morality is not taken to be a causally efficacious part of the world. (Though it might make a difference if people believe in an objective morality, as it might deter them from defecting on the social contract in instances where it would be personally advantageous. Beliefs certainly are causally efficacious.)
2) "History teaches us that social morality changes. Slavery was once good."
No, slavery was once believed to be good. Compare: "History teaches us that social astronomy changes. The universe was once geocentric." This confuses belief with reality. If by "social morality" you simply mean the codes of conduct widely accepted in a society, then "social morality" has very little to do with normative morality, which is what we're interested in here. (See also my posts on society and morality, and especially moral diversity and skepticism.)
3) Subjective moral progress is possible.
In other words, it is possible for the world to become closer to how I want it to be. That's certainly true, but not very interesting. No genuine progress is possible on such a view. The world isn't made better by ridding it of racism and genocide, but merely "more to my taste". After all, racists might think the civil rights movement was positively bad. (Obviously they are mistaken, but we can only recognize this truth if we are moral objectivists.)
4) "How would you determine whether an action was universally moral?"
Well, there's this field called "moral philosophy", where people reason about such things. Some questions are harder than others. Consider the following:
You have the option of whether or not to press a button. If you press the button, then a roomful of innocent children will be subject to torturous agony. If you don't, then nothing happens. There are no other significant differences between the two outcomes. What is the morally best option?
That is not a difficult question. More generally, the moral facts are determined by facts about human welfare, which themselves supervene upon natural facts. (We don't always have access to all these facts, of course, but that doesn't mean they aren't there.)
5) "Suppose there were some way to know that a moral code was universal. Such a recipe would still rely upon subjective moral appeal!"
Yes, we've already granted that human behaviour is a result of psychological facts, not abstract metaphysical ones. See #1 above. Append "So what?"
6) "We have no reason to expect that universal moral laws would be appealing to humans"
On the contrary, I think there is a conceptual connection between morality and human welfare. DL asks us to imagine that moral experts discover that it's immoral to eat tasty food. But unless there is some reason behind the moral claim (e.g. that eating tasty food would somehow cause great harm to others), his scenario is conceptually impossible. You might as well ask us to imagine that baldness has nothing to do with hair.
7) "a universal moral law is nothing without a universal regime for compliance."
So practically-minded! We have social norms and criminal laws to cover such matters; it's not the job of the moral facts themselves to do the policing. Rather, they're there to make true statements (such as "slavery is wrong", and "genuine moral progress is possible") true. That is, we need to posit moral facts for theoretical, not practical, reasons.
8) "Advocates of universal moral law like to argue that we must accept the existence of universal moral truths, or else the things we regard as subjectively immoral would be acceptable. This claim is contradictory because the claim acknowledges the value of subjective morality..."
That's a terrible argument. (To illustrate, one could run an analogous argument with "physical" in place of "moral".) The point is not that whatever I believe must be true, therefore we should posit objective facts to match my arbitrary beliefs. What an absurd suggestion! The claim is instead that some particular beliefs themselves (e.g. "slavery is wrong", etc.) are true, and thus need a matching fact as their truthmaker. Note that it is the content of the belief that makes it true (if indeed it is), not the arbitrary fact of my believing it.
9) "Ironically, it is authoritarianism that poses the greatest threat to our subjective moral good"
I agree. But we shouldn't conflate dogmatism with objectivism. Relativism is not the way to avoid dogmatism. (Quite the opposite, in fact.) I've explained all this before.
10) There's a theistic moral objectivist commenting on DL's post. For the record, I disagree with much of what he says. My post on 'God given value' explains why.
Okay, that's all for now.