Without spoiling, would you say anything hits as hard as the first dungeon? That's mainly what I hear people talk about when they say the game has some rough topics.
Dungeon 4 definitely hits hard but because the content differs from Dungeon 1's story, it may not resonate with everyone.
That's the thing with the other dungeons, they handle different topics and because not everyone has had the same experiences, some of it won't resonate. Dungeon 1's setting is a common experience for a lot of people, making it easier to relate. Additionally, they do a few things in that first story that engage the player a bit more directly adding to that.
Honestly the first chapter of the game is about as bad as it gets in terms of "oh shit, that's horrible" which is kind of a detriment. I mean the subject matter the other dungeons deal with is bad, but there are a number of factors that make the first dungeon seem a lot more personal in a variety of ways.
I don't have a kid, but I don't think there's anything in P5 that a 14+ year old would have trouble dealing with unless they're especially sensitive. Bad stuff is talked about but nothing particularly graphic happens onscreen, and while there are hard-hitting moments, overall the overall tone is somewhere between Persona 3's "EVERYBODY'S GONNA DIE: DEAL WITH IT" and P4's "Zoinks Chie, check out this crazy pantry, maybe there's some steak inside." Although I guess in terms of content, P5 is more about how ugly life can be rather than the ever-creeping omnipresence of mortality.
I actually think that if they had kept doing the rest of the game like how they did Dungeon 1, it'd probably be a much heavier game than it is. It may have even been too heavy. I feel like what they did with the rest of the stories strike that balance between P3 and P4, as you said, that works and still handles a lot of things without bringing the player down.
On the opposite end of the spectrum though, the game has some amazing positive, emotional moments. These moments far surpass anything that P3 and P4 have in their story. I don't want to get into spoilers but I'm sure that folks will agree that the story for the Hierophant Co-Op, in and out of that set of quests is pretty great stuff.
I don't think those are exclusive skills. There are several Persona who have the almighty amps and the baton touch SP recovery at least.
Those are unique skills as Holundrian points out. They don't show up on the base Persona at all, just when you receive them through that function.