Armor-Piercing Question: When Applejack tells Celestia that she didn't want to encourage the CMC to become bullies, she replies that when she was younger, a bully was someone who picked on those who hadn't done anything, and any consequences they suffered was because they'd brought it on themselves.
Celestia: Have things really changed so that just standing up to somepony who's hurting you gets you called a bully?
The Bet: There are two and both involve Twilight and Celestia. 1) Will they reconcile as Teacher and Student? 2) What is Celestia grooming Twilight to be? The latter outcomes involved a combination of 'alicorn', 'princess', or some other authority figure like 'prime minister' or 'head of the royal library'.
Blue and Orange Morality: As explained here, Word of God believes that Celestia is so old, having taken the long perspective and played the long game for so long, that she can't not treat other ponies as pawns, feels that anything not planned years in advance is dangerously spur-of-the-moment, and has forgotten how to let go rather than try to secretly interfere.
The Not My Problem Spell invokes and exploits this so that its caster can move around freely.
Celestia later realizes this fed right into the CMC's bullying issue. Most ponies who witnessed it assumed the parents were already aware and went about their business rather than helping out.
Could Have Avoided This Plot: Twilight lampshades how easily most prior conflicts could have been resolved if Celestia had explained the situation more thoroughly.
Curb-Stomp Battle: With the extra resources Twilight got before setting off for the Crystal Kingdom, Sombra proved MUCH less threatening. Twilight mentions Rainbow's use of this term.
The Ditherer: Spike confides to Cadence that Celestia's constant testing actually fed Twilight's insecurities and made her more like this, always second-guessing herself rather than trusting her own judgment.
Eureka Moment: Celestia realizes that the very nature of the Not My Problem Spell explains why the CMC's bullying problem has gotten so bad.
Fix Fic: Primarily Deconstructs Celestia's Trickster Mentor tendencies by revealing how badly this has hurt her Faithful Student. Also turns a baleful eye on how the CMC-getting-bullied issue has been handled.
Foreshadowing: Celestia's advice about bullies reaping what they sow when their victims fight back becomes a big plot point in The Great Alicorn Hunt when Babs' own bullies show up.
Gambit Roulette: Discussed Trope. Celestia is well aware of how much her plan to redeem her sister depended on the precise interlocking of far-flung events, and she hated it. If she had had her way, she would have found all the Element Bearers years before Nightmare Moon had returned, mentored them in the use of the Elements, fostered a friendship between them, and then directed them to blast Nightmare Moon the instant she returned. Alas, it didn't happen that way.
Idiot Ball: Twilight claims that this is the only reason that they beat Discord; if he had been smart, he would have tossed them into the ocean or into a volcano, not just hid them inside her own house.
Imagine Spot: Cadence has an unnerving one after Twilight points out how Changelings feed off of love, and the Crystal Heart is powered by Love and Hope.
Karma Houdini: Invoked. The tendency for bullies to get away with their actions due to a mentality of "Defending yourself makes you just as bad" and ignoring the issue is brought up and discussed.
Lemony Narrator: The narration occasionally snarks in-story but the trope truly applies in the notes at the end of each chapter where the narrator comments/provides more information.
If anything can match 'banishing my beloved little sister for a thousand years' in Celestia's book of bad decisions, it's 'manipulate my beloved student so much that she severs ties with me.'
As for Applejack, it would probably be 'refusing to see just how badly my little sister has been hurt by bullying even when it's happening right under my nose.'
Mood-Swinger: One does not shake off a lifetime of adoration without side-effects. After Twilight severs ties with Celestia she shifts, rapidly and without warning, from upright confidence to catatonic shock to euphoric happiness to inconsolable depression.
Never Live It Down: In-Universe, Spike still remembers the time Celestia suddenly threw a Fabergé egg at Twilight. Celestia thought it was a cheap decoration and didn't realize it was the real deal until after she had thrown it to Twilight.
Noodle Implements: Celestia briefly mentioned a 300-year plan of hers that was foiled by a tenpenny nail, five scrolls, and a misplaced banana.
When Celestia tries to help Applejack, she makes the mistake of repeatedly assuming AJ did something to help with Babs instead of trying to lecture the CMC after the fact.
Celestia makes the mistake earlier of referring to the situation in the Crystal Empire as a "test," which Twilight calls her out on, as she believes that the possibility of dooming thousands of innocent ponies to hellish fates is not worth "A gold star on a test."
Not So Omniscient After All: Celestia is as fallible as anypony else, and she has made more than her fair share of mistakes over the centuries. She admits to Twilight that her shortsightedness has caused many of her most carefully laid out plans to be thwarted by the most trivial (or occasionally not-so-trivial) of things.
Oblivious Guilt Slinging: When Sweetie Belle agrees with AJ, saying they can't blame her for not knowing how Bloom was being bullied at school... or in town... or at Sugarcube Corner, or in her very own house...
Sweetie Belle: Why shouldn't she not notice something happening right under her nose? Applejack: Layerin' on the sarcasm nice and thick there, aintcha, Sweetie Belle? Sweetie Belle: I was?
One of Us: The story uses many, many tropes by name.
Plot Parallel: Celestia arrives at the train station just as Applejack and the CMC are seeing Babs Seed off, enabling the Princess to listen in on Applebloom calling her sister out and note the parallels.
"The Reason You Suck" Speech: Aside from the abovementioned Rage Against the Mentor, Applejack receives one from the Cutie Mark Crusaders when they point out how, no matter how clear it was that they were being bullied, they never received any help from her.
Rebuilt Pedestal: In the final chapter, Twilight's relationship (and, to a lesser extent, her faith in) Celestia is restored. The princess explains to Twilight why she kept Nightmare Moon's return a secret from Twilight, the real reason why she sent Twilight to Ponyville, and why she sent the Element Bearers on such dangerous mission. She admits her mistake, apologizes, and the two reconcile.
Celestia attempting to use the Crystal Kingdom to test Twilight – and her attempt to defend said decision when called out on it – becomes Twilight's Rage Breaking Point, "because I'm not going to imperil hundreds or even thousands of innocent ponies trying to get a gold sticker from you on a TEST!"
After Twilight leaves, Luna plays with this by stating that the unicorn passed a "true test" by calling her out... and wonders how Celestia will fare with her OWN test of character.
Also Deconstructed when Spike outlines how Celestia's frequent use of these has actually made Twilight less confident about her own abilities. Without the pressure of viewing her trip to the Crystal Kingdom as yet another test, Twilight manages to get FAR more prepared and accomplished.
Selective Obliviousness: Celestia realizes that the reason why the Cutie Mark Crusaders have received such inadequate help in response to bullying can be summed up by this trope.
Applejack had probably seen plenty of signs that Applebloom was being bullied, but she had a farm to run. So she had convinced herself they weren't that important, just 'one of those things foals go through.' Their teacher had probably seen it too, but she had classes to teach, and so she'd convinced herself that posting a few rules on the blackboard about name-calling or fighting fixed it, or that it was too much trouble to tell the victim and the bully apart in a fight, and that punishing everyone – or ignoring it entirely – were much easier.
So Proud of You: Part of Celestia and Twilight's reconciliation is Celestia admitting that she was impressed beyond words when Twilight and her friends defeated Nightmare Moon and became more and more proud of Twilight with each passing adventure.
Spanner in the Works: Celestia briefly mentions a Noodle Incident involving "a plan three hundred years in the making that went completely awry thanks to a tenpenny nail, five scrolls and a misplaced banana...."
Take That: Celestia laughs off the idea that she was going to make Twilight an alicorn. Incidentally, less than a month after the chapter in question was released...
Time for Plan B: What we see in the canon pilot was not Celestia's main plan; it was a contingency. She did not expect Twilight to redeem Nightmare Moon immediately. Instead, without the Elements, Celestia thought Nightmare Moon would defeat her, take over Equestria and usher in Eternal Night, so she sent Twilight to rural Ponyville hoping she would escape Nightmare's wrath and maybe defeat Nightmare Moon in the future. The most likely scenario, in her mind, was her own return from the Sun years from then and defeat Nightmare Moon herself. What we see in the canon pilot was one hundred percent Twilight's own initiative.
Twilight's Rage Against the Mentor is primarily this, though it fluxes into more obvious anger when Celestia pushes her buttons a few more times.
Celestia herself starts getting frosty when Twilight keeps pushing the matter.
Trickster Mentor: Deconstructed. Twilight is so sick and tired of being a pawn in Celestia's games that she renounces her place as Celestia's student.
Undying Loyalty: Even after the viciousRage Against the Mentor, half of Twilight's friends bet money that she would be back in 'Most Faithful Student' mode in short order. It is ultimately a Downplayed Trope; while Twilight is no longer the same adoring fangirl she is still willing to carry out Celestia's assignments and eventually reconciles with her.
Unusually Uninteresting Sight: The Princesses' Not My Problem Spell has this effect, influencing anyone who looks at someone under its effects to decide it's not interesting or worth their time. In fact, it's more effective, the stranger the target appears; thus, they have a variety of strange clothes and accessories for the sole purpose of enhancing its strength. However it can also go horribly right; a footnote mentions a time they went the whole hog and became effectively invisible, which almost caused a carriage to crash into them.
Xanatos Speed Chess: According to Celestia, her various apparent manipulations are more this than anything. She can't plan things with any real certainty, so instead she plans for as many possibilities as she can and uses which plans are appropriate to her needs.
You Were Trying Too Hard: Defied; Celestia did not tell Twilight about being a Bearer of the Elements because she was afraid Twilight would end up trying to force friendships and make the other Bearers reveal themselves.
Notably, it comes back to bite heranyways, as she did not give any explanations for the secrecy after it was no longer needed.