Answering your questions slightly out of order:
We understand that our moral intuitions point to the existence of an objective standard of morality because that is part of the very meaning of those moral intuitions. In much the same was as our subjective perceptions of the external world (though sight, sound, touch, etc) naturally lead us to believe in the objective existence of that external world, our moral intuitions naturally point to an objective standard. I’ve written more on this point on my blog, here.
If, on investigation, we find our moral intuitions don’t match up with the objective standard, the correct thing to do is to adjust our moral intuitions, just as the correct thing to do if our perceptions of the external world fail to match up with reality (due to poor eyesight, for example) is to correct them (through eyeglasses, say).
I’m not going to claim to be the best at articulating them, but here is an example of such an argument, continued in this post and the one after.
The basic reasoning goes like this: objective morality is real (supported by moral intuitions) and if therefore grounded in reality in some way, the possible ways for objective morality to be grounded in reality are A, B, C; B and C don’t work (for such and such reasons) so are false, therefore A is true.
By “nature” I am referring to the essential characteristics of a thing. That God’s character is essential to him is part of what can be inferred by arguments such as the moral argument I linked to.
Because we recognize it as that to which our moral intuitions about goodness point.
Just like Euthyphro, then, this is not a true dilemma. Another option is that God is good and genocide is evil, and God didn’t commit genocide (either because he didn’t do what is claimed, or because what he did isn’t genocide).
But I maintain that this is a distinct objection from Euthyphro, again because it relies on an additional premise. So if you are ready to concede that the original Euthyphro dilemma can be satisfactorily resolved…