Here's a better capture of the context from "The Best of Both Worlds Pt I". In the capture, Riker is presenting the quarters to Shelby for the first time.
PICARD: Number One, why don't you show the Commander to her quarters?
[...much unrelated conversation between Picard and Adm. Hanson about Shelby and Riker...]
SHELBY: I ...
Purely for the sake of completeness, this image is "The Geeky Hallows" by artist TheHookshot.
The objects are:
Link's "Master Sword" (Zelda BOTW, etc)
Gandalf the White's staff (LOTR)
The inscription from the One Ring (LOTR)
Mark IX Sonic Screwdriver (Doctor Who)
Qui-Gon Jinn's Lightsaber (Star Wars)
Harry Potter's (Broken) Wand (or possibly a Kymera Magic ...
EDIT: Adele C. seems to have found the actual language. So go give them some love.
I feel that after pouring over possibly hundreds of fictional languages trying to find an exact match I can confidently say that your friend made the language up on his own. That being said I have spent even more time trying to decipher it and below is what I have so far ...
The Rogue One: The Ultimate Visual Guide factbook describes them as
ILLUMINATED TRAFFIC WANDS
They were part of the outfit that Jyn stole from Imperial "Deck Technician" Kent Deezling. You can see him using them to signal landing ships in the clip below.
This is Space Station Delta from the Japanese series Ultraman Tiga (1996), obviously part of the Ultraman franchise. Specifically, the shot is taken from episode 7, timestamp 1'55:
For the record, it's part of the Terrestrial Peaceable Consortium fleet, the TPC being an organization "ensuring peace on Earth", partly by fighting creatures from outer space ...
From left to right:
The Philosopher's (Sorcerer's) Stone
Tom Riddle's Diary (with inkwell and quill)
The Goblet of Fire
A Prophecy Record
The sign of the Deathly Hallows
Each object seems to represent a book/movie, going in chronological order from The Philosopher's Stone to The Deathly Hallows.
The Brickpedia article on the 2011 Star Wars Advent Calendar says that it is Nute Gunray's walking Mechno-Chair, and seeing it myself, I agree. It was used in The Phantom Menace to talk with the hologram of Darth Sidious while walking.
It looks like the wands used by the air marshaller to direct the cargo shuttles when they were landing. You can see the red sections at the tips of the rods that light up and help indicate to the pilot what they need to do and where to move. Similar to what you see in a real life airport.
Lego version since I can't seem to find the movie version (Valorum ...
Here are all the Star Wars references that I found in the drawing:
Obviously, the main reference is Darth Vader's helmet.
I'm not sure what the thing next to the Death Star is. It could be Alderaan or possibly Yavin or Yavin IV. Yavin IV seems probable.
The igloo in the center is probably Lars moisture farm
The igloo toward the bottom looks like Kenobi's ...
In the original John Berkey painting, it appears that Vader is holding his lightsaber, stylized in a similar manner to Luke's lightsaber. (In fact, it almost looks like a solid metal blade with light glinting off it to me, but I think that's just stylization.)
Berkey's illustrations from early in the history of Star Wars were done with minimal access to ...
In brief, from top-to-bottom:
The Doctor's TARDIS (Doctor Who)
Millennium Falcon (Star Wars)
Battlestar Galactica (Battlestar Galactica)
Saturn V rocket (Apollo missions)
Discovery One (2001: A Space Odyssey)
USS Enterprise (Star Trek)
And once again, with pictures and a bit more commentary:
The Doctor's TARDIS. This is the time machine ...
Those were the rings of Lah'mu, the planet on which Galen Erso was hiding.
Star Wars: Rogue One: The Ultimate Visual Guide
This image from the film gives you a better view. As you can see, the rings themselves are partially shadowed, explaining their abrupt edges.
This is a 9000 Z001 landspeeder
The 9000 Z001 landspeeder was a Ubrikkian model commonly used around
the time of the Battle of Yavin. It was modified from the 9000
landspeeder, which had been aimed at Ugor customers but had flopped.
At the time of filming, the ship was simply labelled (in the construction blueprints) as "Fourth ...
According to the visual dictionary, those 'pens' are actually code cylinders. I'd assume by that they mean it's the authorization codes for that particular officer. (Kind of like a badge/PIN/password combo that you see at secure facilities.)
This Wired article says that the tartan is one of the many Clan Scott designs.
It looks like pieces of the actual tartan were sold and autographed as memorabilia, with a piece recently being auctioned off in 2010, on this online auction site. The page includes a close up of the tartan that allows a better look at the pattern:
From this online registry of ...
After an extensive search, the only reference to these that I managed to find were, as you mentioned, listed as a goof, that simply states
Items on Ben's table as he watches the hologram.
There is no direct reference to these items anywhere else I can find, and even this description is vague, possibly referring to there being 3 items in one shot, then ...
I haven't actually seen this logo used in conjunction with the film, but Aquaman is currently slated to be released on July 27, 2018.
You may notice that the logo resembles Aquaman's belt... buckle?
Click images to enlarge.
In order to fully answer the question I have converted my answer to a community wiki. Please edit as required.
Starfury, Babylon 5 (possibly a Heavy Starfury)
This is a stargate from the Stargate movie and associated TV shows.
In answer to the bonus question about the Stargate. I suspect it is the Earth gate most commonly seen at Stargate Command.
This is not a monument at all, but rather Ra's Mask.
It was actually in the intro for all the first five seasons, intended to be a visual tie-in to the movie, where Ra has a similar mask.
When it moved to Sci-Fi a new intro was made, including for early seasons. This Reddit post explains why some seasons kept the original opening:
The first five seasons, ...