My reasons for assigning ~0% to some of these:
threaten to destroy Azkaban destroying one of the Dementors
They would lock him away to protect the Dementors who are Britain's most powerful magical weapon in reserve in case of war with another magical nation. (Quoth Dumbledore.)
offer the Wizengamot Voldemort's true identity (Quirrel)
Harry doesn't know that fact, so he can't offer it. Anyway, how would it help to reverse the judgement against Hermione?
using the Time-turner to alter Hermione's testimony to something easily falsifiable
Can't use it to change what's already happened. Hermione has already given her testimony, and Harry didn't even listen so he wouldn't be in a good position to subtly modify it. And the Veritaserum on her is already wearing off, precluding further testimony.
using the Time-turner to smuggle Hermione out under the Cloak
Harry and Hermione can't be both under the Cloak at once. People under the Cloak can still be caught by physically feeling around. The Aurors would stop them (certainly the one who wasn't under the Cloak at the time), and if they didn't, Hermione would be running around the building Cloaked but with no real way out.
ask for a trial by combat
Wizengamot would have to vote to make the trial-by-combat's results binding (otherwise why should it reverse the standing Wizengamot vote to punish Hermione?) Lucius will ask them not to vote so, because Dumbledore would be Harry's champion, and so they won't.
More generally, if Dumbledore could challenge Lucius to a duel every time a vote went against him, he'd have total control of the vote outcomes by virtue of being undefeatable in combat. And we know that's not the case.
have Quirrel make Bella confess to the crime
What would be the explanation presented for why Bella comes forward to confess, without implicating Quirrel? Just "Voldemort ordered me to do this and then Obliviated me"? Everyone would suspect Voldemort also false-memory-charmed Bella into believing she did it. And why would Voldemort sacrifice his most trusted and powerful lieutenant, whom he recently rescued at great risk, and not some smaller pawn? And why would Voldemort execute a plan to murder Draco or to frame Hermione in the first place? And how would Bella have gotten into Hogwarts without the wards detecting it, or Dumbledore's Map? And how could she be recovered enough already (if she appears to testify everyone will see she's not very recovered yet)?
And to begin with you'd have to explain to members of the Wizengamot that Voldemort is still alive and that he broke out Bella from Azkaban. Imagine the panic - no reasoned debate would be possible after that. But without telling them this, how to explain that Bella is turning herself in, and that she has been Obliviated of everything during and after her escape?
Oh, and by inspecting Bella they will notice she does not in fact have her Animagus form again, which will cause Dumbledore to rethink some conclusions... (Edit: she has probably regained her Animagus form on Quirrel's instructions, it would be stupid not to. See below.)
Regarding the Cloak, one possibility is that Harry could duplicate it using the Time Turner. (Harry[1] goes back in time, equips himself with Cloak[1], sneaks up to Harry[2] and take Cloak[2] from his pouch. He could use both cloaks to perform an impossible rescue, then return Cloak[2] to Harry[2]'s pouch.)
EDIT: New discussion thread here.
This is a new thread to discuss Eliezer Yudkowsky's Harry Potter and the Methods of Rationality and anything related to it. With two chapters recently the previous thread has very quickly reached 500 comments. The latest chapter as of 17th March 2012 is Ch. 79.
There is now a site dedicated to the story at hpmor.com, which is now the place to go to find the authors notes and all sorts of other goodies. AdeleneDawner has kept an archive of Author's Notes. (This goes up to the notes for chapter 76, and is now not updating. The authors notes from chapter 77 onwards are on hpmor.com.)
The first 5 discussion threads are on the main page under the harry_potter tag. Threads 6 and on (including this one) are in the discussion section using its separate tag system. Also: one, two, three, four, five, six, seven, eight, nine, ten.
As a reminder, it's often useful to start your comment by indicating which chapter you are commenting on.
Spoiler Warning: this thread is full of spoilers. With few exceptions, spoilers for MOR and canon are fair game to post, without warning or rot13. More specifically: