Welp, was going to make a Thread about the missing DLC last year but due to various reasons this got delayed and turned into a full collection of missing/neglected stuff (as GOG was also missing updates)
Given how there's still questions and how explaining HDN is a large tangled web, I'm probably going to do one or two threads soon.
Apologies for this large post, I don't know if I could've made it smaller.

To answer some things, not just to the original posters but also in general: Updated so they handle the new DLC, as a "side effect" we also got the Game Updates (afaik GOG never got a single update after launch)
Japanese's apparently available as Text Language now but needs to be set in the installer.
(On Steam you have Japanese and Chinese, switching via Steam's Properties redownloads like half of the game though Oo)
Not sure if Chinese's available as well.

It wasn't "incomplete", more like it was "overcomplete" with the 3 secret Dengeki DLCs.
When Hyperdimension Neptunia Re;Birth 1 launched in January 2015 it had the following DLC:

Free/DLC
Histy's Trial Item
Histy's Beginner's Item
Fairy Fencer F Collaboration

1.-/DLC
Plutia Battle Entry
Peashy Battle Entry
Histoire Battle Entry
Additional Content 1
Additional Content 2
Additional Content 3

GOG had those as well, only issue was that the Free DLC was bundled with the paid one.
(which still is an issue and was also the case for Re;2 and 3, except there it's worse as the Free DLC is included with the game by default on Steam)

For some reason it had three more DLCs, those were removed with the update now though, more info down below.
(Allowance, DENGEKI Playstation(R) Collaboration Costume Set & DENGEKI Playstation(R) Collaboration Costume Set, Part 2)

On August 24th 2016, the Deluxe Pack fore Re;1 got added on Steam (together with the Deluxe Bundle, which is the game plus this DLC) which apparently was available to Japan since Launch/even as a Preorder.
The Deluxe Pack contains a Digital Artbook, PC Themepack and Soundtrack CD, so no in game content.

In May 2018, Hyperdimension Neptunia Re;Birth 1 Plus got released in Japan for the PS4.
This is essentially a port of Re;1 PC but with additional content.
(Characters from the newer games, adding some things to address issues like grinding)
It seems like IFI doesn't seem to plan to release Re;1+ in the West at the moment, but will release all the addons as individual DLC for the PC Version.
(Vita owners seem to be left empty handed while PS4 owners will probably have to wait a good while, not sure if Sony's Censorship put a hold to this)

As such, the individual upgrades have been sold as DLC , starting on the 05.09.2018 with the first batch (Tag Along, AV Club and Colosseum + Characters DLC)
While this was just for Re;1 initially, there have been DLC releases for all three games starting from the 16.04.2019

GOG finally got those in the second DLC Pack now.
It's worth noting that if my theory is correct, there will be even more DLC for Re;1 in the future.
I also expect Re;2 and 3 to get Plus remakes (and as such more DLC content) as the Vita's possibly nearing end of life, with it being the only way for Console Players to play Re;1-3.
I do question if the Survival Mode DLC works though as it's linked to a Steam Leaderboard and only rewards the top scoring players with items (terrible idea)
I switched over to Steam as well as I expected them to neglect this (asked a year ago and got told that Retroism did this as well as that they have no interest in this) as well as Steam having the Spinoff Titles.
Unfortunately, the Spinoff Titles, bar one, were done with other Japanese Developers. The one that wasn't is a crossover with characters that are the property of SEGA, and we all know their stance on DRM Free Stuff (they want to marry Denuvo at this point)

It's worth noting that the DLC is sold individually (as in each character/addon, although some like the Colosseum + Characters DLC exist) while GOG had them in Packs.
The question about the Story was already answered, although for completionist's sake, I'll list it here:
Re;1 is in a different Dimension and as such isn't linked to Re;2 and onward (although it's still a good game to start as the character personalities are more or less the same across the universe)
Re;2,3 & VIIR are a continuity, the main dimension is the same, although you'll be traveling between different Dimensions.
VII/VIIR isn't the roman numeral 7 but V II/V 2, a continuation of Victory/Re;3 V Generation.
Spinoffs all take place in their own Dimension.
While the continuity exists, I've heard that the games don't make too much use out of it, so one could just start out with any of them.


Since the cat's out of the bag and this would be noticed sooner or later:
Yes, this seems to be a mistake, oddly enough those were the first three new DLCs released for Re;1.
4 (Characters following you in Dungeons, feature from VIIR) and 5 (view cutscenes again (Possible in all games after Re;1 from what I've heard)) are kind of things that should've been part of a update, although I doubt that they meant to include those here :P
You should be able to deactivate them in the DLC menu though, you'll have to scroll down as the list is a lot longer now.


It's worth noting that the DLC display is a bit odd now.

Before this update, DLC would show up as "Purchase Required" if you didn't have it.
Installing it automatically enabled the DLC, giving you a on/off toggle.
Setting the DLC to off made it show up as "Purchase Required" again, requiring you to manually edit the Neptune.ini file to renable it.
From what I know on Steam the game "phones home" to Steam to check if you own the DLC and then gives you the toggle ability.

Now, all DLC gets shown as on (won't actually work without the files though) and you manually need to toggle them off (once done it's saved in Neptune.ini though)

*DENGEKI, not DEDENKI :)
I was able to confirm the same (does GOG offer any way to get the old DLC installers?)
The Dengeki Content does still work when manually added to the DLC Folder or via the old DLC installer but only shows a empty message, just like you said.

Regarding the "DENGEKI Playstation(R) Collaboration Costume Set, Part 2", the general assumption seems to be that the Costumes for Neptune are the same.
This isn't the case, they're actually different, the only difference are her panties though.
Given how Re;1 doesn't have a "Free Camera" Plan like Re;2 and 3 do, you can rarely see that though (Character is KO'd and laying on the ground, certain attack animation frames)

I don't know where those three DLCs (Allowance, DENGEKI Playstation(R) Collaboration Costume Set, DENGEKI Playstation(R) Collaboration Costume Set, Part 2) come from though.
I've checked Japanese Re;1 on Steam as well as the Vita Store (since the game was originally on Vita), no info either.
Seems like even Japanese players on imageboards were confused about this DLC (and unhappy about the Region Lock)
The only guess I could make is that those were exclusive to the Dengeki Magazine
(as Neptunia U apparently had a code there, some Dengeki Items were also released as a DLC for Super Neptunia RPG on Steam, the latest spinoff title)

It's worth noting that if you go by the DLC folder numbers, Neptunia Re;Birth 2 also has a missing DLC (9) while Re;Birth 3 has several (4,10,11,12,13)
I wasn't able to find info about Re;Birth 2 but found the following for 3 (Translated via Google Translate) In short, it looks like we weren't supposed to have those, yet they slipped past.
Personal opinion, I don't think that something that's been out for nearly 5 years now (28.01.2015) should be removed.

Sorry - I am an absolute noob in these things and so I did not notice that I was constantly spelling this wrong. :)

Btw: +1 for the great and dtailed explanation.

No Problem, it's not a word I see and use every day as well :P
And thanks, unfortunately had to make two posts and kinda worry that it's a wall of text that nobody wants to read.
(actually couldn't send the whole thing, GOG's Message Window kept loading... so I had to post it in 4 parts (two of which got merged together))

On a side note, it's pretty nice that we finally got updates as those addressed some old issues.
Previously, most weapons wouldn't have a sound on use in the Dungeon, now the Hammers, Spears, Broadswords and so on do, looks like all sound effects, except the default swing were missing (Battles use different SFX than the Dungeon ones)

Your post was awesome, +1 from me too, I am all for informative walls of text.