I believe it was (first?) mentioned in the Animate pamphlets from 2020*, but Anne's pet bird also came up again in the character trivia shorts they posted in the leadup to REVOLUTION! It's a Java sparrow named Tamago.

(*I had to piece translations for profile stuff of out a bunch of apparently-now-deleted tweets, photos, and privatters, so the part that mentions Anne's sparrow never made the cut because I could never personally verify the phrasing.
I'd love to eventually revise my translations of this material but I don't feel like paying $4 per pamphlet for five year old sheets of paper that mostly says stuff I already know, lol.)

I am going to lead with the fact that this is alleged, based on vibes, and 100% conjecture, and I am not trying to cast any aspersions.

But yes, the impression I get is of machine translation that is lightly overseen or edited by a human, probably by an in-house Japanese L1/English L2 speaker. This is (fairly verifiably) what they use on the site for profiles et al, so I feel pretty confident that it's probably what's going on with the official lyric subtitles.
Song lyrics inherently force you to infer a lot of context and struggle with line-to-line cohesion, which they sometimes struggle with, and you sometimes get lines that read very awkwardly (a few of Kanata's lines in Instinct really stand out to me and are fresh in the mind, but Akan Yatsura are always a nightmare).

I don't love that it's there, as a renowned MTL disliker, but I'm also not super pressed that a primarily-Japanese-facing company is going with the least labor-intensive option when it comes to things they have to handle purely in-house for a niche and tertiary market.

But that's part of why I retranslate the songs, lol. (And because I enjoy it.)

(Interestingly, I think that the official translations of videos that they post on Facebook -- e.g. drama track previews and the birthday videos from last year -- smack definitely more of "a human got properly paid to do this", and this makes me a little more generous about the fact that the song lyrics that go up on the Japanese-facing channel are wonky, since when they're actually English-facing they do seem to have more QC.)

Thank you for your kind words!

cozmez unfortunately have completely evaded me in terms of a concrete explanation like most of the other groups! I do have a page where I give my thoughts on all the name origins of things (https://phanto.moe/other/handles), but to skip to and lightly rephrase the cozmez part:

It might be meant to evoke Gozu and Mezu, the ox- and horse-headed twin guards of the gates of hell. This would kind of track with their general vibe, but feels a little flimsy IMO.
It also might be meant to evoke the word 'cosmic', which seems more plausible now with them leaning hard into the space/alien aesthetic, but note that the aliens were not introduced to their aesthetic until Exhibition Show so this probably wasn't the original intention.

So there's never been an official statement about this and it is in some ways contradicted by other bits of canon. So there are two explanations, and I will lead with the one I think is the most "real" first.

1, which is the out of character answer: It's just a stylistic thing.

A big thing about Paradox Live's art is that the official art (directed by Planarico) takes stylistic choices by the designing artists and canonizes them; you can see this most notably in how cozmez's white pupils and Iori's red pupils are directly lifted from (respectively) Kinako and Harada's stylistic choices in the original character designs. Aki Akane drew Hajun's hair very pale in the cover art for the Exhibition Show CD, and the art team probably just liked it and canonized it into the baseline style going forward (even though Aki Akane's later art for anniversaries has used Hajun's original more yellow hair color, but that art is less "canon" than the Exhib show art).
It also may be to help differentiate him visually from Dongha, though that is an afterthought since there was a full year between Exhib show and Dongha's introduction.

2, which is the explanation some people use to explain it in-universe: I've seen some people say that Hajun's phantometal erosion episode may have lightened his hair.
This is not fully substantiated by canon (Itsuki still has dark hair despite prolonged exposure to phantometal and Haruomi's hair going white is because he was visibly aged), but if you really need an in-character explanation, you can use that as a headcanon, since phantometal's effects are pretty flexible and as-the-plot-demands.

It appears that the birthday commemorative posts for the upcoming year is going to be cheki polaroids, yes, and the cycle always begins with cozmez for logistics reasons*.
The fact that they are specifically cheki plays into the "characters as in-universe performers" thing that they're going for a lot harder with BoU, what with the characters doing photoshoots for album covers etc.
(And, idk, man, people were mad that they did mini-dramas last year instead of getting new art, so, www, you can't win em all.)

(*They had the first birthday in December 2019, after Paralive was first announced in November 2019, and so they're first up in the birthday rotation; any birthdays between the anniversary and December 24th were later additions, and so didn't get put into the rotation until a year had passed.)

Currently the word is that it shouldn't affect anything; GCrest is just no longer their own subsidiary (GCREST), and are instead a studio (GCRESTUDIO) under Sumzap, both of which were/are under the CyberAgent corporation; so it's basically a corporate consolidation.
All we really know is that the things they've announced with their franchises are still going forward as announced.

Now, to be clear, I don't know much about corporate mergers, especially Japanese ones, but at the moment we don't know enough to say whether this will affect anything other than some nomenclature or if it may indicate any more major changes. My bet is on "nothing will change except for what company's name shows up in the publisher credits" -- it's not as though GCREST were independent and got bought out, they were already under CyberAgent -- but until we start seeing actual dramatic shifts I'm not going to try to predict the whims of a bunch of Japanese businessmen.

So for the moment, with what we know now: I don't expect anything to change just yet. My opinion may change when we see more or if we see the effects of the consolidation, but I'm not expecting anything super drastic for a hot minute at least.