999 isn't a super-complicated game, but it assumes you have a lot of intuition, and a bunch of things go unexplained. This is a primary text analysis that teases out some answers from the text we actually have.
Let me reiterate: this is a primary text analysis, meaning an analysis of the text itself. This discounts extra-textual sources like author interviews and sequels - 999 was originally a standalone game, and while it had a stinger, it wasn't clear there was ever going to be a sequel in the form of Virtue's Last Reward. Obviously, later works (most notably Zero Time Dilemma) retconned some of 999's text, but that's beyond the scope of this analysis.

If you're reading this anywhere but Steam, you're reading stolen work.

This deserves to be talked about first because it's a radical departure from a major plot point in the original game. On the original 999 (for the Nintendo DS), the top screen showed 'adventure mode' and the bottom screen showed 'novel mode', which are later revealed to be Junpei and past-Akane's points-of-view respectively, which is why they sometimes address things in different ways. This means the player is, for most of the game, de facto playing as past-Akane, providing information to Junpei to help with the game; at the end of the game, the most obvious way to solve the puzzle (it's a different puzzle in the Steam release due to being for a home computer) is to turn the DS upside-down - you are now acting as Junpei, sending information back to past-Akane.
The single-screen nature of this release somewhat reduces the power of this plot twist, but you still get most of it across.

June is a potential present-day copy of Kurashiki Akane. Note that in Japanese her codename is more significant: Murasaki literally means 'purple', but the 'mu' part can be read as 'six' or as 'nothingness', alluding to her role as Zero. (Her taking the codename 'purple' also alludes to her being related but separate from Kurashiki Aoi and past-Akane, whose names are derived from 'blue' and 'red' respectively.)
The idea that she's only a 'potential' is suggested by her status on routes that move away from the true ending. Her fever flares up on those routes, and on the Safe/Zero Lost ending, June simply falls faint and disappears when Junpei takes his eyes off her for a moment, as the potential to save her in the past no longer exists, so June in the present can't exist, either.
Conversely, as long as the possibility for past-Akane to be saved exists, present-June exists too.
Further, it's clear that June understands the position she's in, and has set up clues for past-Akane, whom present-June knows is seeing through Junpei's eyes: a mandatory clue in Junpei's starting room is taped to a mirror, so that past-Akane can see who she's attached to, and present-June's schizophrenic behavior is, at least in part, putting on an act for the benefit of past-Akane to reassure her that things will be okay. In particular, some of the stories June shares, like the one about Stead (more on him later), take on a different tone with the benefit of this knowledge: she's assuring past-Akane that what she's experiencing has happened before and she's not losing her marbles.

This is probably the most difficult question to answer, because it's very likely that Zero's identity retroactively changed. Santa/Aoi notes that he is 'Zero's assistant', and he and June between them strongly imply that she is Zero (past-Akane says it explicitly), but present-June only comes into existence once the possibility of saving past-Akane is on the table.
The most likely answer for who the original mastermind is is actually Seven. By his own admission, Seven has spent the intervening time between Nonary Games pursuing Hongou and Cradle Pharmaceuticals, and studying the Nonary Game. That would have lead him to the information and resources necessary to stage the second Nonary Game in an attempt to save past-Akane, but that's a Herculean task for one man, and so he went to someone he already knew and who would agree with his wild plan: Kurashiki Aoi.
However, at some point, present-June would come into existence as a possibility; while the possibility of past-Akane being saved exists, present-June not only exists, but has always existed. Once this happens, it seems June took over Seven's place as the mastermind, not only during the game, but in the creation of the game as well. This places both June and Seven in quantum superpositions, and Seven's memory loss is heavily implied to be a symptom of being in this state where it's not clear what actually happened in the incinerator. Junpei actually twigs to this in the ending when he spots Seven looking satisfied, but decides to keep his mouth shut about it.
Of course, Santa/Aoi's "3" bracelet is really the 0 bracelet, so in a sense, he's just as much Zero as present-June and Seven are. He is also likely the Zero who speaks with Junpei at the end of the Zero Lost route.
Speaking of Seven's involvement, it's Seven who is the one that suggests they take up codenames, on the basis that they shouldn't let Zero know anything about them (to which Santa agrees). The characters (and the reader) go with this at the time, but if you stop and think about it, that doesn't make sense: Zero knows who everyone is, because Zero ambushed and captured everyone in the first place. What it does do, however, is protect people's identities from Hongou (except for Snake, who is outed on account of his blindness; this is likely why Zero chooses to replace him during the sabotaged REDs sequence).

Putting the water level aside, once you start hearing about the first Nonary Game, it's actually pretty obvious that you're in Building Q if you think about it: Building Q housed the 'transmitters', who had the clues to the puzzles, and the Gigantic housed the 'receivers', who had to rely on receiving the solutions from the transmitters in order to escape. However, the puzzle rooms you solve have clues in them - so you're in the 'transmitter' part of the experiment.

Likely was not behind a numbered door in the original Nonary Game, and instead served as an escape vessel for the gamemasters. Presumably, it would have been part of a new-game-specific puzzle, like the number cards puzzle in the Sokoban room, since the submarine wouldn't actually have anywhere to go.

The number cards puzzle in the Sokoban room can't be solved by someone with prosopagnosia (at least, not in a timely fashion). Not coincidentally, solving this puzzle is required to get the golden gun, an item that almost certainly did not exist in the original Nonary Game, as it would add noise to the experiment. (Although this isn't super well thought-out, which is actually true of a number of things in the new Nonary Game, and as a consequence Ace ends up with it in most routes.) However, the number cards puzzle has a secondary benefit: anyone who went there with Ace would find out that he couldn't distinguish faces. This is likely why June has a noisy fall in the Sokoban room - she knows Junpei will rush over to check on her, but it's not because of her fever, and Santa supports her assertion that she just tripped. Doing this causes Junpei to discover that Ace is unable to solve the puzzle, which is a key in uncovering his treachery on the Zero Lost route, along with finding out that the murdered man in Snake's clothes is not actually Snake.
Speaking of Ace, Zero's offer to let him go free isn't a false one: if you tally up Ace, the Ninth Man, and the two other Cradle Pharmaceuticals members who Zero inserted into the game, they would have been able to leave through the exit q door together. However, Zero correctly anticipated that Ace would whack all three of them, which ultimately leads to his downfall.

Although the game only allows you to reach the true ending by first seeing the Zero Lost ending, in-universe, it would have been possible to complete the Nonary Game without passing information across timelines. Zero sets it up so that opening Snake's coffin is mandatory to get enough players after the Ninth Man is killed, but the "truth had gone" note that Clover has on the Zero Lost route isn't something she wrote herself - she found it on Cap's corpse when she took his bracelet, but she only mentions it to Junpei on the Axe ending route, where she brings it up to cover that she stole the bracelet and fire axe. Presumably, on the true ending, it's actually still in her pocket (as she still has the bracelet), but the characters never figure it out. This is particularly weird because Clover's death in the Zero Lost route implies that she figured out that it was the code to safe, which is part of why she went to the 1st Class Cabins and had the note in hand when Ace stabbed her.

Although the group identify the 0 bracelet Cap is wearing as being a number 6, it's actually more likely to be the letter O and represents the number 24 (O=24 in base-35). For math reasons this will result in being functionally equivalent to a 6 in most cases for digital root purposes, but it's interesting that nobody mentions it, even though Snake explicitly suggests it might be a letter O.

Unlike every other player in the game, Lotus has no direct connection to the previous Nonary Game, only a secondary one through her children. Her inclusion is consequently rather unusual, but there are hints about why she might have been included:
Being a hacker, Lotus knows a lot about computers and numbers. Since there are a lot of number puzzles included, she's like a cheat code for the characters, and her teaching the others hex saves them in a few puzzles.
However, there's a secondary reason that she might have been included - to provide closure. Lotus never found out from her children what happened when they disappeared in the first Nonary Game, and it obviously haunts her; while she has no direct connection to the Nonary Game, it still affected her, and Zero may have decided to include her on that basis. Zero's secondary goal was, if not to save past-Akane, to force Hongou to admit his crimes and broadcast it around the world; doing so would have given Lotus the information she was searching for.

When Snake explains the 4-leaf clover symbolism to the kids in the first Nonary Game, he mentions that his sister's name is Clover, even though Clover takes that on a codename in this Nonary Game. This makes sense in Japanese: her given name is クローバー kuroovaa (Clover), but the codename she takes is 'Yotsuba', which refers to a plant with four leaves on one stalk (i.e. a four-leaf clover). This is a pretty poor deception, but it manages to deceive Ace because of his inability to attach a visual identity to her, which is all that matters.

William Thomas Stead[en.wikipedia.org] was a pioneer of investigative journalism who did some absolutely insane stuff, was an advocate for "Government by Journalism", and conducted a sting investigation into child trafficking in London that made the paper he wrote in, the Pall Mall Gazette, sell out instantly, with used copies trading hands for 20 times the original price and 10,000 members of the public basically besieging the Gazette's headquarters.
He also had often claimed throughout his life that would die either by lynching or drowning, and published two pieces (one in 1886 and one in 1892), the former about a ship-to-ship collision of an 'unsinkable' ship due to a lack of lifeboats, and the latter about a vessel named Majestic that rescues survivors of an iceberg-induced shipwreck.
Stead was on the Titanic's ill-fated voyage, and the 'automatic writing' detail that June uses him for (and which Stead claimed he used to communicate with his assistant editor, Ada Goodrich Freer) isn't the only thing he contributed to 999: apparently, during the 11-course meal, he entertained his dining companions with a variety of tall tales, including a tale that the unlucky mummy was present aboardship (it was not) - this is the real-life anecdote that links Stead, the Titanic, and the mummy together.
Apparently, after the impact, he helped women and children into lifeboats and gave his life jacket to another passenger; according to an eyewitness, he had clung to a raft, but "(his) feet became frozen" and "(he was) compelled to release (his) hold".
June brings up Stead's 'automatic writing' as one of her fun occult facts, but with the context of knowing that past-Akane is seeing this, the conversation takes on a different light: she implies that Stead accessed the morphogenetic field and sent information back in time to himself, leading to his publishing those pieces. Presumably, in this context, this means Stead boarded the Titanic despite knowing his fate in the hopes that he would be able to send more information back to himself, repeating until he managed to arrive at a timeline where the Titanic took more precautions, had enough life boats, and so on. (This is actually something of a can of worms, as a successful voyage would make it impossible for Stead to send information back in time, meaning nobody would be warned. Presumably, this is why the second Nonary Game has the secondary goal of seeking revenge on Ace: it provides a way for June-as-Zero to retroactively construct the game, even though past-Akane has always been saved while present-June exists.)

The story that you hear a couple of times about a substance no longer being producable, or only being a producable form, is actually a real one! (However, "EDT" refers to a different kind of chemical in real life, and isn't an industrial cleaner like Seven claims.) Wikipedia has an article about this.[en.wikipedia.org] I looked it up thinking "no shot this is a real thing" and it sure is. I like this a lot, actually; most of the pseudoscience or rumors that are brought up in 999 are references to real pseudoscientific theories or pre-existing fiction, but often with an extra twist thrown in that muddies the water for the player. Similarly, Lotus' 'brain as output device' discussion is based on a theory that was gaining some popularity in east Asia around the time 999 was made, called the "Whole-Body Soul Theory" (and like a million other things), which basically suggests the same principle, that the ineffable essence that we might sum up as a soul is contributed to by a whole, healthy body, and the brain simply renders that into an output in physical reality (in the form of neuron impulses etc.); this 'explains' why people's personalities often undergo a change if they lose limbs or organs. (It was popular in Japan, which generally doesn't believe that trauma is real, so the explanation is basically a way to invalidate people's traumatic experiences and say they're no longer a complete person! Sasuga Japan!)

In the true ending route, when everyone converges at the incinerator, June blinks out of existence when Junpei stops looking at her for a moment due to Seven's severe headache. During this period, Junpei has briefly lost his connection with past-Akane, and it seems like the puzzle will not be activated, which cuts off the possibility of present-June existing. However, once Junpei reestablishes contact with past-Akane and saves her, the other characters remark that June left with Santa and Ace. Junpei's memory is apparently immune to this ripple effect, as he doesn't know where June is (and we the player don't see her leave with them). This suggests that after past-Akane was saved, present-June ceased to be in a superposition and was permanently, retroactively added to the event, revealing herself as Zero alongside Santa and then leaving with Ace in tow.

That's everything I've got for you, and it should answer most of the overhanging questions.
Let me reiterate again that this relies on the primary source, which is to say it's an analysis of the text itself, discounting extra-textual sources like interviews with the authors and retcons in later texts (most notably Zero Time Dilemma, but VLR does a little bit of this too). So, while these explorations are satisfactory as answers based on the text itself, they may not be canonical in terms of the wider Zero Escape franchise.