1.  To whom does the Stone of the Hapless refer?  Why did the Valar preserve this memorial?  Why would they preserve the memorial and not the people themselves?

The fate of Húrin's family is an exemplar of the fate of the Eldar, ground to (almost) nothing by the might of Morgoth (and their pride in daring to challenge him) and perhaps by extension, the fate of us all, fighting the long defeat.

2. Are Mim’s claims valid?

We don't know enough about the Petty-Dwarves and their history, really.  They apparently were not a separate family of dwarves, like the Longbeards, but an exiled group.  Did they earn that exile?  Why did the Sindar hunt them?  Had they abandoned Nargothrond, or did Thingol or Finrod kick them out?  And what happened to Mîm's son, Ibun, by the way?  He was captured with his father before the betrayal of Amon Rudh, and he's apparently dead now (if Mîm is the last of his kind) but we never learned his fate.

3.  Why did Hurin kill Mim?  Did Mim deserve to die?

"I daresay he did," as Gandalf would say, but Húrin would be better not playing executioner.  How about a "lost tale" of Mîm's life?

4. Why was "one thing only" taken by Hurin?

He's not in it for the money--he's got things he means to say.  It's a princely gift, to be sure.

”For now my fate is fulfilled, and the purpose of Morgoth achieved...”

5. To what event(s) does now refer?  and  6.  Was Hurin more despondent upon learning the truth than he was believing Morgoth’s lies?  and  7.  What happened to Aure entuluva?

I love this scene, Húrin at last understanding and quietly, sadly going his way.  Morgoth turned Húrin's proud mockery to bitterness, but at last he's free.

Does he know what he's done, bringing the necklace to Doriath, planting the seeds of Thingol's doom?  Or does he think his final unknowing service for Morgoth was just his harsh words to the king?  I don't know.

As for Aure entuluva, he knows, not for him.  It's a little like Lear, brought low by his pride, seeing the truth, but too late.  Did Húrin actually throw himself into the sea, by the way?  We are only told that, "It is said" that he did so.  Who said it, and why?