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EDIT: Accidentally duplicated post. Please see https://startrek.website/post/25896182 for all responses and put future responses there. Also, I have more theorizing there.
For the first part, why he doesn't have a formal education: Nog is living on a space station with his uncle and father. Not in a Ferengi society. The Uncle doesn't want to shell out for an education for the kid cause he's Ferengi and doesn't want to pay for things, and the father just goes along because he's kind of a doormat. There's not enough opportunity for job-shadowing that allows Nog to learn from a Ferengi mentor that's capable of teaching such things as literacy, so he remains ignorant. Also explains why he gets into so much trouble (aside from the bad influence of his uncle), as he has too much free time from lack of mentorship.
He also shows that he's very hard working when he applies himself. So, in addition to tutoring from Jake and probably Keiko, he just buckles down and does the work. Especially when he gets to the academy. He specifies in "Homefront" (Or "Paradise Lost", I don't remember exactly) that he's doing all right at the Academy. Not excelling, so he's struggling. Probably not the least of which is because he's behind in education as compared to his classmates.
Nog went up the ranks more quickly because of the war, which provided him with many opportunities. Also, his connections, which allowed him to intern on DS9 and the Defiant, again in the context of war.
Harry was on a ship isolated from Starfleet. If Janeway had promoted everyone who had the merit, she'd have had a ship full of commanders and a handful of captains by the time they got home. But most of those people would have had to keep doing the same jobs, anyway.
But the true reason for the discrepancy noted by OP is just down to the fact that Voyager's writing team just wasn't as good as DS9's (or they were held back from above). The only characters to have a meaningful arc on Voyager were the Doctor, and Seven. The others mostly just had to reset at the end of every episode to preserve the show's episodic nature.
Harry never got promoted because the writers never figured out how to evolve his function on the show. Now, I assign a small part of the blame to Garrett Wang himself: he rarely gave performances that stood out and made the writers want to write about him more. But most of the blame goes to the writers and show-runners.
For example, if they'd exploited dual crews more throughout the series, with more conflict, etc., Harry might have found a place. Imagine if the Maquis crew had had their own ship for a big part of the series? Harry, eventually frustrated at being passed over for promotion, might have joined them!
Harry never got promoted because the writers never figured out how to evolve his function on the show.
This one is a bit of a copout, because Kim's official role as the ship's operations officer would absolutely have been appropriate for a higher ranking officer. It's the same job Data held as a Lieutenant Commander on the Enterprise; if anything, the strange bit is that it was given to a green ensign in the first place.
Ultimately, the real explanation is a much sillier bit of bad writing. According to Garret Wang, quoted here:
Kim was probed, beaten, tortured and held the distinction of being the first Voyager crew member to die and come back to life. What more does a guy have to do to get promoted to Lieutenant for frak’s sake? To add further insult to injury, other crew members such as Tuvok (Russ) and Paris were being promoted, demoted and then re- promoted throughout the seven-year run of Voyager.
I’m not trying to be negative here; just saying it like it is. During the fourth season, I called writer/producer Brannon Braga and asked him why my character hadn’t received a promotion yet. His response? “Well, somebody’s gotta be the ensign.” Geez, thanks. Thanks for nothing.
Why it was important that "somebody’s gotta be the ensign" is a mystery to me.
Kinda surprised they hadn't made Naomi Wildman an ensign by the end of their run.
Hard work, dedication, and Lobes.
Harry Kim got addicted to clarinet
Maybe Janeway was just secretly really racist against Asian people. Just because society has left bigotry behind doesn't mean all members of that society have.
Janeway kept Kim on a permanent coffee-finding mission.
You can’t delete the duplicate?
The problem is people commented on it already.
Rip