The reason Harry never questioned why he turned into a dragon during primary school? None of the muggles around him noticed anything different, so Harry assumed becoming a dragon was just something that happened to people sometimes.
Harry is confused by Hogwarts "await(ing) his owl" and wonders if they have an owl shortage, and if so why they're asking for owl donations from the students.
Since Harry isn't responding to his Hogwarts letters, the school starts sending him more at an exponential rate. They still don't explain what's going on, so Harry takes to hoarding (and eating) them. By the time letters start pouring through the chimney, he has enough to make an inch-thick layer at the bottom of his cupboard and is wondering if he can fold them into origami.
When Hagrid shows up at the Dursley's house, Harry is in the middle of a dream about an old teacher of his running a charity book sale to buy replacement owls.
While on the train to Hogwarts, Harry gets asked the same few questions about himself so frequently, he starts mentally referring to them as 'the obvious five questions'. They are, in order:
"Are you really Harry Potter?" (Yes.) "Why are you a dragon?" (Don't know, sorry, it was a while ago.) "How did you stop You-Know-Who?" (I was a baby, I don't know that either.) "Can we see the scar?" (It isn't as if I'm wearing anything in the way) "What magic can you do?" (None yet, that's why I'm going to school.)
Harry asks Professor Sinistra why Astronomy is part of the core curriculum at Hogwarts. After going over how celestial bodies factor into other areas of magic, Sinistra tells him why she thinks it's a core class: Dumbledore likes astronomy. Apparently, this is also the reason why Hogwarts has a Divination class and clubs for Dumbledore's hobbies of chamber music and ten-pin bowling.
Peter Pettigrew's cover is blown because the Smith twins enter the wrong dorm while trying to prank the Weasley twins and catch him reading Redwall in rat form.
Third Year
Sirus shows up to help Lupin administer the obstacle course section of the end-of-year DADA exam. Seamus asks if his mother's claim that Sirius bribed his way out of Azkaban prison is true; Sirius is unamused and says that on an unrelated note, Seamus is going first.
Fourth Year
Knowing that he's a master of Loophole Abuse, Fred and George turn to Dean when looking for ways to get around the Age Line protecting the Goblet of Fire. The only problem turns out to be that Dumbledore asked him first.
Subsequently, when the pair of them end up in the infirmary with beards, it's noted that the Age Line had had the "opposite effect" to prove its point to the Smith twins, who are pointed out as a pair of disgruntled fox kits in a basket in the corner.
Fifth Year
After Dumbledore gives him a demonstration of alchemy and Harry finds it interesting, the dragon comments that more people would be interested in learning about the subject if Dumbledore showed them how to do it. Dumbledore then realize he's completely forgotten to mention the existence of his NEWT-level Alchemy class for the past several years.
Post-Astronomy OWL exam, Harry and his classmates discuss what a "T" grade (the worst possible) would even look like as applied to a star chart, which soon extends to talking about hypothetical T grades in other subjects. Dean Thomas suggests that a T grade in Care of Magical Creatures would involve getting mauled by a flobberworm, which is an entirely harmless creature; when Harry asks how that would even be possible, Dean comments that pulling it off would probably exceed expectations (and thus get an E grade).
Sirius relates that when he did his OWLs he tried and failed to spell "ADOPTED" with his grades.
Sixth Year
Sirius tells Harry that his friend Remus didn't take NEWT-level Astronomy because the practical part of the final exam was on the full moon, thus Remus wouldn't be able to attend due to being a wolf that night. Harry jokes about the exam being held on the full Moon (rather than the day of it) and comments that he thought Ron was trying to be the first wizard to go there; Sirius makes "a strangled noise" and wonders how he didn't see that joke coming. Harry also has a vivid mental image of Remus attending an Astronomy NEWT in wolf form and calmly writing RAAARGH on a chart.
A first-year manticore ends up in Hufflepuff. Colin Creevy complains about how none of the students with lion bits have been Sorted into Gryffindor (represented by a lion) yet.
McGonagall is confused by a first-year named James Tuckett being on the Sorting list, as a student with the exact same name started attending Hogwarts three years earlier. Not helping matters is that the new James Tuckett is sorted into the same House as the first one. The Sorting Hat actually apologizes for the confusion.
Professor McGonagall tells Harry's friends about Dumbledore forgetting to tell people about his Alchemy class. Dean says that that sort of thing is "peak Dumbledore."
Like in canon, Luna is chosen to commentate the first Quidditch match of the year and her observations continue to be... odd:
Luna mentions that while Gryffindor and Slytherin have played several times before, "talking about those previous games is useless" because their line-ups have changed since.
Luna wonders aloud if she's supposed to introduce the teams (presumably the players) when they enter the pitch, then refers to the participants as the Slytherin and Gryffindor teams without actually naming anyone. Harry and Hermione conclude that this still counts as introducing them.
After Ron blocks a shot on-goal by kicking it up the pitch, Luna wonders if Muggles have a game with a similar premise. Dean is left questioning whether she's joking or legitimately doesn't know about Association Football.
An hour and a half into the game, Luna starts talking about special Snitches that are invisible and made of platinum. McGonagall asks her to please stay on-topic to the game and that they've already discussed this; Luna serenely replies that it is on-topic due to being Quidditch-related.
Seventh Year
Voldemort's attempt to mug Harry during the summer holidays ends with him blowing himself up again using the Killing Curse. Sirius bursts into laughter when he learns about it, and the whole thing is such an Epic Fail that the Daily Prophet decides that Voldemort was an impostor instead of the real deal. One of its readers goes so far to suggest that the fact it failed so badly was ample enough proof that it was the man himself.
Epilogue
A year after the Battle of Hogwarts, the Astronomy Tower has been repaired. However, there are occasions where it rains on the Tower, and only the Tower, and the skies are still clear when it rains there.
While Ron is planning out his spaceship, Dean criticizes how wobbly his linework is. Ron comments that the lines would be straight if he used a ruler, but it'd also take two hours to draw; he then says that he heard Harry's stifled giggle as a response.
Dean Thomas comments that it's going to be weird for Harry's Defence Against the Dark Arts students to have the same professor for their entire time at Hogwarts. Harry agrees and comments that he should wear different hats.
Harry's recounting of 2017's incoming first-year students lists the two new Weasleys separately from the other humans. He also has no idea how Fred and George didn't have kids at the same time, as said Weasleys are Fred's eldest son and Ron's eldest daughter.
