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?
It is a memorial to the suffering and the fight against Morgoth, so it is
fitting that it should be chosen to remain standig after the destruction of
Beleriand.
Are Mim’s claims valid?
In a manner of speaking, yes. The Petty-dwarves are like the Native
American tribes who were displaced when the more "advanced" civilization came
and renamed all the land.
Why did Hurin kill Mim? Did Mim deserve to die?
This is a tricky moral question, and one which doesn't have an easy
answer. Mim did betray Turin, and Turin had made amends with him for
earlier wrongs. Does he deserve death, though? Punishment yes, but
Hurin's family lives in a world governed by vengeance and ancient codes of
honor, so the killing of Mim is consistent with the cultre we are
presented. This chapter, though, vividly shows the consequences of
vengeance and of living by the sword.
Why was "one thing only" taken by Hurin?
It was most prized by Finrod and is a token of the special relationship between
Hurin's family and the fallen king.
”For now my fate is fulfilled, and the purpose of Morgoth achieved; but I am
his thrall no longer.” To what event(s) does now refer?
Hurin has now paid the "debt" he owes Thingol for the hospitality and aid he
extended to his son, wife, and daughter. This act, in a traditional
society, settles the score, or is a token of Hurin's gratitude (and it makes up
for the undeserved anger he expressed earlier). Now Hurin is able to die
with no "loose ends."
What happened to Aure entuluva?
Hurin in the end might not be able to gain comfort from hope eternal, but the
future generations will be able to see the day come again. For now, he
has laid down his arms in the
fight.

Howe, Drowning of Númenor