There are several Omakes on other ways Harry's Sorting could have gone. They all seem rather silly until you remember how that Hat works according to HPMOR: it borrows intelligence and memory from the wearer. Those things definitely sounded like the king of pranks Harry would do. The "Headmaster" thing instead reflects his egomania rather than his questionable humor.
In Chapter 10, everything Harry says to convince the Sorting Hat he should be sorted "where I belong" does exactly that, only Harry thinks he belongs in Ravenclaw, but all his arguments about not achieving his potential elsewhere prove to the Hat that Harry really belongs in Slytherin.
In chapter 18, Snape correctly answers Harry's question on how many valence electrons a carbon atom has. Why would a wizard know this relatively obscure bit of information? Oh right, he's a Legilimens! (Or a Muggleborn. Half-blood Prince, remember? He could easily have read that out of a textbook and remembered it.)
Quirrell seems to be taking his teaching duties seriously, marking boring homework assignments and advising Hermione to leave Hogwarts for her own safety (though there might be other motives for that). This seems at odds with his cynical, unconcerned attitude, until you realise there was a time in canon that Voldemort wanted to be a Defence Professor.
This one also has additional explanations in MoR-canon:
Quirrel: I will tell you what does not hold my interest... Grading Ministry-mandated essays does not hold my interest, Mr. Potter. But I have undertaken the position of Defense Professor at Hogwarts, and I will see it through to its end.
Quirrelmort is pretty much an immortal Nietzschean Übermensch. Until he hears a prophecy he does not like and resolves to tear it apart at every possible point of intervention, he's pretty much just doing stuff for shiggles, because he chooses to (and to mentor his young apprentice/younger self).
And even after that, Hogwarts is the place to be to do so, since it lets him be close to Harry and be - physically - in the best possible place to intervene if he tries doing anything stupid, as well as to see it coming by interacting with him on a nearly day-to-day basis. Remember, you can't apparate into Hogwarts...
Also, just about everything he does as a Professor does further that goal - mentoring Harry does so since having Harry as his delegate and Friendly Enemy or ally will increase his effectiveness, at the very least by freeing up his time from ruling Britain. Teaching effective Battle Magic, as well as various other thinking skills, proactivity, and how to effectively fight in an army also increases the combat-readiness of his students, to fight the Muggles, as Quirrel believes they will have to. He is also preparing them to accept the leadership of Harry and/or Draco. So by that logic, teaching at Hogwarts is an investment in the future. He also, of course, wanted the Philosopher's Stone.
The bloke's also known for the effectiveness and thoroughness with which he plays roles, and unlike canon, his patience. He's chosen to take on the role of Hogwarts Defence Professor, and so:
Professor Quirrel: Not everyone who teaches here has been the best, but the best have all taught at Hogwarts. In such august company, and after so much time anticipating this day, I would be ashamed to set myself any standard lower than perfection. And so I do intend that every one of you will always remember this year as the best Defence class that you have ever had.
Moody thinks rather than using a trapped Triwizard Trophy to teleport a victim (with a completely pointless return charm on it), an intelligent dark wizard would have someone simply toss the target a trapped Knut on their next visit to town. Quirrel used this exact method to take Harry to Diagon Alley (albeit consensually) earlier in the story, making this a call back (and reminder of Methods!Quirrel's intelligence compared to canon Voldemort).
In Chapter 74, the students who encourage Pansy Parkinson to think her soul has been eaten are all members of the Chaos Legion.
Quirrel's answer to Dumbledore after Harry killed a dementor makes us wonder why nobody caused him any problems after a Sarcastic Confession of this sort. Until we realize that the only people who would take this answer at its face value are those who know the answer to the "What lies at the center of a Dementor?" riddle, and those who had weren't likely to find Quirrel's goal in life to be all that bad.
On the same note, what everyone except Harry hears the Dementor say to Professor Quirrell is that "it knew [him], and that it would hunt [him] down someday, wherever [he] tried to hide." Harry assumes this is just something they've all constructed, rather than something the Dementor has actually said; whether or not this is true, the reader knows there's more meaning to that threat than the characters assume.
In the story, We learn that "normal" patronuses are reflections of the casters' not thinking about Death. Now, Dumbledore has a phoenix patronus. Why is that? Dumbledore's way of escaping death is that he strongly believes in an afterlife… and the Phoenix (a being who is reborn whenever he dies) is a metaphor for that.
In Chapter 89, all the chracters who try to stop Harry are from the Chaos Legion, and those who help him are from the Sunshine Regiment
Daphne's lightsaber is a Blade of Greengrass
Dumbledore's "there would be nothing left of the world but fire!" in chapter 61 is what Severus is thinking of when Rianne says "they would catch fire and burn the cauldron" in chapter 76.
The Interdict is a very good reason to make people immortal.
In chapter 74, Quirrell describes an old and terrible ritual, one intended to summon up Death - involving a noose that has hanged a man, and a blade used to kill a woman. He mentions offhandedly that the ritual to counter this has been lost to time. This sounds remarkably like the cause of the Dementors...
Crosses with Fridge Horror, in Chapter 103, Quirrel gives Hermione a D on a Ministry exam, and laments that he would have given her a lower grade, but that would have been "in poor taste". Considering how, in canon, the only grade lower than a D is a T... for troll...
Harry worries that the Line will be intercepted by a subtly evil heir, but this will not happen if Harry himself succeeds in living forever, or hands it to Hermione when he fails to as she is almost guaranteed to live forever. Unless this is the one where Harry himself turns evil, in which case everyone is screwed.
Minerva's presence did make a final difference to how the prophecy turned out. If it had been just Severus there with Trelawney, obviously he would have figured out that it was meant for him, and that would have required it to be even more ambiguous and subtle to make him still go to the Dark Lord with it before working out what it meant. (Also, if Harry's logic is correct, it's the only reason she is included in the fellowship and thus there were many interactions with Harry that would not have happened the way they did.)
Chapter 109: when Voldemort regains his faculties the mirror summons Dumbledore. Why? Because Voldemort's deepest desire is for a worthy opponent.
At the end, when Voldemort summons the Death Eaters, he doesn't call them by their proper names - rather, Mr. White, Mr. Grim, et cetera. This is a very definite Call-Back to chapter 7 where Harry requests that Ron call him 'Mr. Spoo', and they eventually settle on 'Mr. Bronze'.
Sirius is called Mr Grim because he can (or could in canon) turn into a big dog that looks like the omen called "the Grim" in Divination.
The rest of the world thinks Dumbledore was waiting for the most dramatic moment to fight Grindelwald; Dumbledore reveals that the Holocaust was powering Grindelwald and he had to stop that first. Why the difference? Because most wizards don't think Muggles are people worth protecting or knowing about, and Dumbledore is an unusual exception!
One of Canon!Harry's favourite spells is Expelliarmus; the disarming charm. He uses it regularly throughout the canon, and in the final fight with Canon!Voldemort, the spell is what wins the fight. In chapter 114, How does Harry defeat Voldemort? He literally disarms him. Admittedly it only extends to just below the wrists, but it still counts.
Subtle play of words. Throughout her trial, Hermione's alleged crime is referred to as a " "cold-blooded murder". Now, remember the alleged murder was committed by gradually cooling the victim's blood.
'DO NOT MESS WITH TIME'. Why is this the end result of any attempts at causing a paradox? Because in order to resolve it, the universe - and by extension the Time-Turner - is trying to determine the simplest way of creating such a stable loop, without computing a thousand different results first.
Unknown if planned by Dumbledore, but all the Canon!Trio's "standard" solutions for the Third-floor corridor obstacles seem to resemble Quirrellmort's specific talents! He would do great against Fluffy, because he has mastered to sing a lullaby, as seen in Hermione's trial arc. He could pass through Devil's Snare by relaxing, as he was trained to lose in an Asian monastery. He could catch a flying key, as he enchanted his bones. He could win a chess game as he had the outstanding mind of the strategist. He would have easily defeated a boggart as he was a Defence professor, familiar with Riddikulus charm. And then there was a potion room, containing a riddle.
If horcruxes work the same way as in canon, then someone has been horribly murdered while in the presence of the Pioneer plaque and the deed has been covered up afterwards.
When Draco is briefly forced to consider siding with Dumbledore, he decides he'd rather be burned alive. And then when you read it again with the knowledge that Draco believes Dumbledore killed his mother by burning her alive while she slept, that comparison becomes rather horrifying. And very similar to the decision that the Potters took — dying rather than letting Voldemort kill Harry.
The story indicates that teachers can freely use memory charms on students without oversight or appeal (after the ritual incident, Snape modifies over a dozen students' memories without the Headmaster's knowledge). While we haven't seen any actual abuses of this, there have been tales of Defence professors who really get around or are more Obviously Evil than Quirrel, and if, like Snape, they can do it without Dumbledore's knowledge, the implications are not pleasant.
If Harry's theory about how Fred and George pranked Rita Skeeter is correct, then Fred and George were able to False Memory Charm her and get away with it. The scale of the pranks they (or less honorable people) could effect with that method are frightening. Which, in hindsight, actually makes it a bit heartwarming that they erased the knowledge of how they did it.
When Hermione calls Quirrell evil: "Miss Granger," Professor Quirrell said gravely, "it can be dangerous to give people compliments like that when they have not been truly earned. The recipient might feel bashful and undeserving and want to do something worthy of your praise...." And what happens in the next arc? Hermione dies horribly. Coincidence?
If The Reveal in chapter 102 of the true functionality of the Horcrux spellis true, and the Pioneer Plaque is indeed a Horcrux then Voldemort has successfully fucked the Plaque's message of peace and introduced all of humanity as assholes to any aliens who find it, as whichever alien picks it up will get Mind Raped. Nice Job Breaking It, Hero!.
Bellatrix got splinched. Evidence: her severed arm still moves, and her Dark Mark was still active enough to be used to call others. She was probably off somewhere feeling still connected to it, blood flowing from her body to it and back. Imagine how she must have felt when it got incinerated.
The extent of how dangerous magic can become, as mentioned in chapter 119. Attempts to quantify the risk of just transmutation on forums or reddit have ranged from the civilization-ending (a few nuclear weapons or biological weapons) to the world-ending (a black hole) to tearing apart stars within several hundred light-years (transmuting an ultra-dense volume of charged subatomic particles like packed electrons) to universe-destroying (violating colour symmetry by transfiguring a single red quark, transfiguring up a lower vacuum state). Given that this same society considers locally suspending the Newtownian laws as an afternoon's work, or what the hidden law of potion making might mean to someone who really understands what makes up iron or uranium, it's quite probable someone might mess around with these aspects by accident even before working with transfiguration.
All it might take was one single Muggleborn who knew the names of the six quarks deciding to try it. That could be the clock ticking down to the prophesied end of the world.
The "DO NOT MESS WITH TIME" note must have come from a Harry who created a different timeline and experienced something horrifying.