Compared to other lines in the Lunaverse theme song, Raindrop's line ("Stormy Weather") seems out of place and overly negative. But this reaction misses the entire point of My Little Pony: Friendship is Magic: that people can disagree with you and you can still be friends with them anyway. For Raindrops, stormy weather is happiness, and a good friend would understand that and let them use that as her contribution to the music.
Luna became Nightmare Moon because the ponies didn't seem to appreciate her night. In other words, Luna wanted to connect but couldn't. Celestia, on the other hand, became Corona because she tried to rule every aspect of her ponies' lives in order to keep them safe, and in the process she destroyed the connections she previously possessed. This may explain the difference in their reactions to the Elements. Luna, even as Nightmare Moon, wanted the love, respect, and friendship of the common pony on some level, and that may have left her open to the healing powers of Harmonic magic in a way that Corona, who had deliberately severed every bond of friendship she possessed, could never be.
This might also explain the attitude of Ponies in the Lunaverse versus their canon counterparts. Lunaverse ponies tend towards more suspicion than their counterparts... which makes sense when you consider the lessons learned by the Betrayal. Luna's betrayal was out of a desire for more friendship and companionship, which serves the narrative that ponies are optimists who see the best in people. Everyone having a bad day could be a good person! Corona, on the other hand, deliberately severed those bonds to become a tyrant. She did so out of a misguided belief that she knew best, even at the cost of the most powerful magic in this setting. Though she was defeated by Luna, the ponies learned their lesson here that everyone has darkness inside of them. In a way, it's the pony version of Original Sin.
It's mentioned in "Crisis On Two Equestrias" that even the most powerful Enchantment spells fall apart if you try to do something violently contrary to the target's nature (like trying to make a mother ignore her child being put in danger). This little tidbit helps explain why Thrash Metail's brainwashing music didn't really work on Octavia. Octavia is devoted to music, devoted to it on a level virtually no other pony is, devoted to it like a mother is devoted to her child. For a spell to try and make her believe that bad music is good contradicts her most fundamental nature, and even the most powerful Enchantment couldn't pull that off.
Also, something similar happened before when Lyra encountered the Sirens- their music couldn't affect her because it was off-key, and like Octavia, Lyra's special talent is music.
Dinky's cutie mark talent (at least in this universe) is making friends. Her mother is one of the Elements of Harmony. Like mother, like daughter?
It's mentioned that in the Lunaverse, Celestia handled the Crystal Empire thing a thousand years ago via orbital strike with a solar flare. Given that the Crystal Heart helps regulate emotions in Equestria, and it was presumably destroyed, that might account for why the Lunaverse has more people indulging in negative emotions than in canon and damaged the racial psyche of the ponies.
Unicorn magic still follows the same rules even for alicorns. Even Luna and Celestia have a much harder time casting spells not related to their cutie mark, it's just that they are so powerful that even the fraction of their full strength they can bring to bear on spells outside their mark is still several times what the most powerful unicorn could muster. But when they raise the sun and moon, they're casting their cutie mark spells, and they can bring their full power to bear.
This also explains why Corona defaults to sunfire for her big attacks. It's a spell directly related to her mark, so it's a lot more efficient than other forms of attack she could use.
One of the big ways the Celestiaverse differs from the Lunaverse was that, in the Celestiaverse, Nightmare Moon was all but forgotten, whereas fear of Corona is one of the foundations of Lunaverse society. A second difference is that, thanks to L!Luna's warning, C!Celestia knew that her sister's imprisonment would end. The two just might be interconnected. C!Celestia knew that her sister would be released, knew that there was a good possibility that the Elements would restore her mind, and knew that if that happened, she would need to be re-integrated into Equestrian society. It is therefore quite likely that she deliberately suppressed the memories of Nightmare Moon, relegated them to a children's fable, so as to minimize the backlash when Luna resumed her rightful place.
For that matter, this also offers yet another possible explanation for the difference between Luna and Corona's reactions to the Elements. It may well be that Corona was not redeemed because the Elements could not imagine the bare possibility that she could be redeemed. The Celestiaverse Elements feared Nightmare Moon, but they didn't have that kind of bone-deep, almost instinctive rejection of her that is such a feature of the Lunaverse.
According to Word of God, all mortal ponies in the Lunaverse are descended from six alicorns who sacrificed their immortality to render themselves capable of procreation. This explains the bizzare way pony genetics interact: since all ponies are descended from that single source, they are fundamentally one race even when divided among the three tribes.
A thousand years in the sun and Celestia's madness hasn't really ebbed at all. But it's not surprising, even if alicorns don't go mad from isolation (Per Word of Double Dash), the last thing she saw before she was banished was her own sister corrupted by Tirek's dark magic, so Celestia has had to spend a thousand years stuck in the sun, utterly convinced her sister had been warped into a monstrous perversion of herself, unable to do a thing about it.
It's been stated and shown in various stories (and elaborated on by RDD) that L!Luna and C!Celestia have rather different styles of governance. Luna favors consolidation of power (fewer ponies with lots of responsibility) and delegating a lot of her authority to her subordinates. Celestia favors bureaucracy (lots of ponies with limited responsibility), and holding the ultimate authority herself. While some of this is undoubtedly due to personal styles, it may also have to do with the different crises the two have faced.
One of Luna's defining traits is her perpetual fear of following her sister into madness and her guilt at effectively abandoning Equestria for over a decade immediately after Corona's defeat. Therefore, she has built her government so that if she ever disappears (whether through falling into madness and being defeated, or through falling into grief and abandoning her nation), the Night Court can still run without her.
Celestia, on the other hand, has deliberately built things so that nopony outside of herself has real power or authority. Thus reflecting both her trauma from her sister going mad with power and the desire to take on sole responsibility for everypony's well-being that consumed Corona.
The comics have stated that Luna's transformation into Nightmare Moon in the Celestiaverse was partially due to the influence of some kind of alien presence (the Nightmare), and this strongly implies that one of the reasons that she was able to make a HeelFace Turn so easily after the Elements defeated her in the first episodes was that the Elements removed the Nightmare's corrupting influence. If this is true, however, then Corona's refusal to accept Luna's offer of forgiveness implies that she was not being influenced by any outside force, and became the Tyrant Sun entirely on her own.
Worse, as more is learned about her character, we find that she's a zealot- she believes she's doing the right thing.
As bad as the fiasco with the mislabeled Zebra rituals was, Lyra and Trixie are incredibly lucky that it wasn't worse. As Trixie herself points out, she could have quite easily killed or seriously and permanently harmed Lyra by unknowingly casting what amounts to totally random spells on her.
What's more, she's lucky that the creature she turned into was capable of surviving in Equestria's environment. Imagine if she turned into a creature like a Tube Worm, which requires living next to a volcanic vent under miles of ocean water? Or a Volus, which require a methane atmosphere and super pressurized suits so that they don't explode?