Does one have to be aware that one is exercising one's free will, in order to have free will?

Hard to see why, in my opinion. If (say) a free action is one that you undertake such that, at the moment of acting, it was at least logically, and perhaps even physically, possible that you either perform that action or not perform that action, those facts themselves at least seem to be independent of what your awareness is. What would be interesting is an argument that shows that only IF one is aware of the facts just described could those facts obtain ... but at the moment I don't see how to generate such an argument. Perhaps in the mix here is the thought in the other direction, a kind of old-fashioned argument for free will, that states that if one believes one is acting freely then one IS acting freely--or using our conscious experience of (or as of) acting in a way in which it seems to us that multiple options are logically and perhaps physically available as a sufficient condition for acting freely. (Your question concerned whether such awareness was a necessary condition, but here it is offered...

Compatiblism is attractive because it finds room for human freedom in a deterministic world. But objections that compatiblism is evasive or incoherent strike me as persuasive. Setting aside the indeterministic defense of free will, how might the hard determist endorse the claim that humans generally do bear moral responsibility for their actions? Or would the hard determinist have to bite this bullet and conclude that moral responsibility is illusory if we have no free will?

I like Stephen's answer, but I think you ARE asking about the hard determinist -- you're convinced by hard determinism about free will (i.e. tht determinism rules out freedom, not (directly) that it rules out moral responsibility), and you're worried about having to give up moral responsibility. But I suppose a lot rides on how one defines moral responsibility -- I don't think it's somehow intrinsic to the concept that we have to have freedom in the indeterminist way in order to be morally responsible. Dennett, for example, makes pretty powerful arguments that we don't and really shouldn't care about whether we 'could have done otherwise' generally speaking -- see Elbow Room. Now you could use that point to defend compatibilist accounts of freedom -- we're free despite determinism's being true -- and then hold that moral responsibility requires freedom. But since you are persuaded against compatibilism (though I hope you've read Dennett...), why exactly couldn't you hold that moral resopnsibility does...

Why doesn't consciousness defeat the determinism argument? If a person consciously decides to order a hamburger instead of a cheeseburger the next time he goes to a restaurant, what force is controlling him to delude himself?

One typical way of thinking of such examples is this: perhaps the sequence of conscious mental states we enjoy is a causal sequence, so "causation" would be the "force" you are asking about. Perhaps the purely determinist laws of neuroscience dictate her sequence of brain states, which in turn dictate her sequence of mental states, generating her "conscious decision process" by which she eventually concludes she will order a hamburger. Perhaps event he deterministic sequence generates/dictates all the "feelings" she feels to, including the feeling of compelte freedom from external forces ... After all we are NOT typically aware of what is causing our mental states, are we? So even our "feeling of freedom," our "sense" of controlling our thoughts and decision processes, may be generated by entirely deterministic causal networks .... It seems to me, then, that consciousness could not itself defeat the determinism argument because our sequence of conscious states could easily be deterministically...

Is religion the true enemy of freedom in a democratic society since it teaches us that we have to think a certain way or is science since it teaches us that nobody is truly free but a product of deterministic forces?

Or another mode of reply: First suppose that science DOES suggest determinism. How would anything be different in our lives? Wouldn't democratic processes work precisely the same way as they have been? (After all, our behavior has been deterministic all along, so why would discovering/proving/merely believing that it is deterministic change anything?) Or since 'freedom' seems to be the larger concern for you, again, what would be different? All the cases where we've held people responsible for their behaviors, we still would hold them, wouldn't we? we'd still lock up bad people, teach our children to be good, etc.... So it isn't clear to me why scientific results would threaten anything, really. Ditto for religion: if we think religions are in the business of generating true claims about the world, then, where they succeed, we should be happy to endorse their claims (assuming we want the truth). Whichever dogmatic religions you're thinking of ARE dogmatic because they believe they have the truth...