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There was an official crossover poster for Gurren Lagann Otoko & Kill La Kill, by the artist who did the Otoko manga.
(For reference, this was back in 2015)
 
Kazuki Nakashima says he first got the idea for piercing the Heavens and dimensions when he was hanging out his laundry.
(Funnily enough, Kiyal first sees the heaven-piercing final battle when she's hanging out her laundry.)
 
Gridman Universe is infinite in size, and thus a mere human merging with him risks loosing themselves forever.



 
Good news and bad news.

Good news; the Transformers special movie collaboration between Trigger, Transformers and the band Bump of Chicken (who also did the opening to Delicious in Dungeon) is out now! Akira Amemiya is the directer (as it should be), with all animation being lovingly hand drawn. The song featured is Mayday.

Bad news; it's not yet available in my country...or anyone elses country.
 
Although it's not yet been officially translated into English, there have been fan translations (thus, a big thank you to the user Savaralyn on Reddit for translating these)! I will also have a look at the official translations when they eventually come out. Of course, these include spoilers for late in the series (and season 2) so be wary...

The Quetzacoatl is 3km long, thus making it the biggest monster, capable of swallowing any other monster in a single gulp. IT possesses overwhelming power, and can travel through the land, sea and skies at incredible speeds.
The Winged Lion is confirmed to have come from an infinite world (thus, the entity of the infinite world should at least be universal+).
 
Otoko is set in a 'present day parody multiverse.'
A story set in the present day parody multiverse where Kamina, Simon, Yoko, and Nia from Gurren Lagann are living happily together.
 
Good news! I have tracked down yet another Gurren Lagann book! The Groundwork of Gurren Lagann: The Lights in the Sky is a book comprising the different keyframes for The Lights in the Sky are Stars, and in itself doesn't have much in the ways of lore. It does, however, contain a few snippets of importance. The Anti-Spiral is said to emerge from the rift of the Mugan Gate.
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From the Rift of the Mugan Gate
 
In episode 4 of Gurren Lagann, a mysterious figure can be seen at the very start (is this LordGenome right after the Anti-Spiral battl? Kamina's dad? Someone else?).
 
Reuploading a very important scan; the Computer World is the world which exists inside computers, and has a causal relationship with the real world (something affecting one will affect the other); the Computer World is comprised of a different world for every computer, and passroutes connect all of these computer worlds together; although the Computer World can be thought of as virtual, entities from the Computer World possess mass. Akane create her world in a corner of the Computer World.
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I have rewatched Gurren Lagann at long last! Of course, I picked up a few new things and remembered some old things (such as either LordGenome or Tepplin powering all of the beastmen's gunmen, and with the Spiral Kings defeat all of the beastman gunmen stop working), and a lot of stuff from pre-timeskip gets overlooked (for instance, the appearence of a galaxy during the Giga Drill Break may be due to Spiral Power being drawn from galaxies in other dimensions).

Enki's special attack is a heat ray with enough power to wipe out a mountain.
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When Viral confronts Gurren Lagann in the throne room, not only had he been made immortal, but Enkidu had also been upgraded to a new four-armed form called Enkidudu. However, even this form was still effortlessly defeated.
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More confirmation that Enkidu was effortlessly defeated.
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01, Enkidu was easily defeated by Gurren Lagabb. The power he gained from the Spiral King was not intended to defeat humans.
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Enkidu
Viral's Gunmen, Enkidu further modified to have four arms and four swords, was instantly killed by Gurren Lagann, despite his confidence.
Dai-Gunkai's crew includes various sea-themed beastmen, including Captain Octa the octopus beastman, Unico the sea urchin beastman (beastwoman?), Ambry the anglerfish beastman and more.
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Dai-Gunkai crew
Captain Octa, nickname for the octopus beastman, Unico, the sea urchin beastman, Ambry, the anglerfish beastman, and more. A full lineup of sea-themed beastmen.
 
Happy Birthday to Hiroyuki Imaishi! Thank you for making so many masterpieces, and I hope you continue to brighten our lives by making so many more.
 
Masaru Sakamoto (the character designer for Gridman Universe) made a poster for Transformers One.
Masaru Sakamoto did an interview on this poster (in Japanese).
 
Akira Amemiya drew the Trans-Homers for Transformers 40th anniversary.
 
Are there any quotes on the Anti-Spiral controlling the dimensionality of his universe?
All I recall is the one where the Anti-Spiral 'transcend space and time and control the multiple dimensions' from Final Drill (which isn't the same as controlling how many dimensions there are).
 
The screenshots of the artefacts found in the Horologium Chamber are broken (again, thanks a lot Imgur), but thankfully the LWA wiki recorded all of the flavour texts, so thank you!

The Calamity Rod has the power to destroy the world, and witches seeking its power sought to control it with a spell of binding.
A rod capable of destroying the entire world. Witches long ago sought to control it with a spell of binding.
The Judgement Rod is used by a messenger of the gods to carry out their, and those with evil harts dare not touch it.
A rod used by a messenger of the gods to mete out their judgement. The evil of heart dare not touch it.
The Arcana Rod can draw back the veils of time, and a skilled witch can use it to gaze many centuries into the future.
A rod that draws back the veil of time. A skilled witch can use it to gaze many centuries into the future.
The Stardust Rod has power over the very stars in the sky, and shooting stars may be caused by thsi rod.
A rod with power over the very stars in the sky. If you see a shooting star, this might rod may be at work.
The Soul Wand feeds on the wielders soul (though it leaves no permanent mark).
A wand that feeds on the wielder's soul. It leaves no lasting mark, but it still sends shivers up the spine.
The Wand of Yggdrasil was carved from Yggdrasil itself, and its power has only since grown since it's creation.
A wand carved from the trunk of a massive tree. The power within has only grown since its creation.
The Fate Wand is powerful enough to twist fate itself.
A wand with power great enough to twist fate itself. Few make the sacrifices needed to master such magic.
The Vision Wand tests those who wield it with an illusion, and those who break free of this illusion can wield the wand.
A wand that tests those who would wield it with an illusion. Those who see through it can take up the wand.
The Dragon Wand is implied to contain a dragon within it, and those without the power to wield it are devoured by 'the beast within' (also as an aside, one needs to have the power to wield a wand of power, and not just in this example).
A wand shaped like a roaring dragon. Those without the power to wield it are devoured by the beast within.
The Chaos Wand possesses devastatic magic, and can amplify the power of the wielder unendingly.
A wand with devastating magic befitting to its name. It can amplify the power of the wielder unendingly...
The Light Wand shines with brilliant light strong enough to pierce any darkness.
A wand that flashes with brilliant light strong enough to pierce any darkness.