Battle Moon Wars/WMG

Kohaku's medicines (and food) are a way of using her Synchronizer abilities without traditional Intimate Healing.
This is not to say they are placebos, but she embues them with her own essence (blood?) radically increasing their potency. The act of needle penetration facilitates the act of connection.
 * This aspect of her ability was possibly awakened by becoming Saber's Master. Since her magic is literally her body; being connected to Avalon just takes it further. Avalon healing her health, and her health being her magic that powers Saber creates a massive feedback loop that makes her able to feed Saber mana via a Magus Ley Line without Deus Sex Machina or the more traditional years of training such a Ley Line normally requires.
 * Which in retrospect, is awakening more of her magical abilities; which leads to...

Kohaku is a Wild Mage.
That's right. Magical Amber is real. Somehow, without the Church and the Mage's Association realizing it; Kohaku developed reality-nudging powers; her obsession combined with her complex medical knowledge; combined with exposure to Tatari mutating into.....Kohaku. Kohaku's "mecha-robots" are actually very low-level mana Dolls. There's been some spillover to Hisui via Twin Telepathy; at least so Hisui can fight off vampires with a mop.
 * Being a doll master, Tohko recognized it right away, but she sees no reason to volunteer information.
 * Thus the secret character Mecha-Hisui that can join is the result of Kohaku "exercising" her skills; this model is far stronger because Kohaku's magic has become stronger. This is also why Mecha-Hisui can summon the Stagnation Spirits when guarding Avalon.
 * Stage 53 is the full breakout point to Awakening.

Stage 53 is the results of an inversed Soul Compression
So the Soulpression Ritual was made by sacrificing the Masters and Servants who fell to the faux-Grail War and stuffing the Master's (and Angra Mainyu's) Mana into the land of Misaki city; then powering the ritual through the cobbled together Tatari/Grail fusion. This resulted in the mana of the land spilling out in the form of some weird spirits. The Servants souls returning would open the path to the Root where the Devil was waiting and could then be nullified by the trigger release of the Soulpression Ritual caused by the Grail/Tatari killing off the entire city. But what happened instead was that right before the Grail/Tatari was going to be activated; Kuu was defeated and the WerkNasu crew just beat the crap out of the Devil instead with the help of Ilya.

However, the groundwork for the ritual was all set up. All that mana had to go somewhere; and the Battle Masteries sacrificed the spirits instead of human souls; effectively putting the process in reverse. So it returned to what resembled its source the most. A person who had the power of the Grail (Saber and a piece of the Grail); and the power of the Tatari through a magic doll that was created under it's influence. (Mech-Hisui). This massive glut of mana flooded Kohaku and Saber; and both transfigured them (into Amber and Lily) and multiplied the Tatari-related Mech-Hisui one hundred fold. This was a bit of a strain on Kohaku's already unique grasp of reality and the weirdness of the Tohno Family Takeover Plan occurred. (The mana backlash occurs when Saber notes her connection to Kohaku mysteriously weakened for a moment; but returned when she became Lily) This was controlled via the events of stage 53 though, and Kohaku became able to manage her Amber persona and abilities.
 * With this new power, she will be able to sustain Saber indefinitely even without Sakura and the Grail fragments.
 * If this is true, this also means she's effectively immortal. Well, she can still die; but as long as her connection exists with Saber; Avalon will prevent them from aging. Rin does comment in the original Fate Stay Night that some mages have achieved symbiotic immortality through their Servants from previous Wars.

The Tohno Family Takeover Plan was Kohaku testing Saber
The context that Kohaku gave for Saber to stay with her after the Tatari Grail War was slightly flimsy, especially for Kohaku. So she let her Id go screaming out; in full force, to see if this would drive Saber away. Saber stayed at her side; slightly worried about what she was doing but more worried about Kohaku's change in character.
 * Remember, Kohaku won. Even though the NasuWerk crew destroyed the Mecha-Hisui's and defeated Amber, Lily and all the Arcueids, the knockout amnesia gas incapacitated them. Kohaku could have done whatever she wanted at that point; but she put everything back instead.

Saber and Kohaku are going to win a future Grail War
Obviously not the Fuyuki Grail, since that Ritual was corrupt and is going to be dismantled completely by Waver Velvet. (There's going to be a lot more gunk in the shreds left this, though.)

The Devil was Erebus.
The continual Malice of Humanity; the Nasu Gang; with the help of the living Grail; bring out it's "heart" and drive it away. It will take a while to be able to reform into a presence again. It's not dead; note that even with all of the special attacks on its Heart; it's HP wasn't reduced to zero; even with the special effects explosion.

Shin'i is an undercover branch of the Time/Space Administrative Bureau.
Earth was producing way too many mages to go ignored, but to declare the planet as "Administrated" would be disastrous; criminals from the Administrated Worlds would flock to the abundance of Logia technology; while nobody wants the various vampires, demons, Church Militant, native mages and other "yokai" to spill out into The Multiverse. So a few Familiars and agents were sent to rustle up and arm a native force with Magitek to deal with things; or at the very least to give an early alarm whistle to other worlds should things be about to blow. Takumi and Haruna probably don't know about Shin'i's origins; but Mike the talking house cat and Natsuki probably do.
 * Vampires, robots, and spirits already exist on Earth in Triangle Heart 3 Sweet Songs Forever. Not a stretch.
 * So far, they haven't been able to have too much of a foothold due to Earth's native weirdness being too entrenched.
 * They'll probably be keeping an eye on Shirou from now on, considering that he was able to create an intelligent device quite literally out of thin air.
 * ...the fact that there's now actually going to be a Magical Girl Lyrical Nanoha Vi Vid / Fate Kaleid Liner Prisma Ilya Crossover makes this especially fitting.

Aoko tied Avalon and Saber to Kohaku
Whether part of some master plan or else For The Lulz (with Aoko, it's hard to tell the difference), this was to get both sides tied together. She does tend to be the Wizard Who Did It and she did manipulate to get Touko and Ryogi involved.
 * She likely got ahold of Avalon through swiping Shirou's original body after his consciousness was transferred to Touko's doll.

Ilya is the new Tatari
We know in Actress Again that just killing the Tatari Core (Oberon, aka the Night of Wallachia) doesn't make it go away, the mechanism is still there. In Actress Again all it took was blood that Oberon had consumed but not vampirized to gain sentience and take over the system. After the clusterkerfluffle that was the Soul Pression ceremony, there's a much stronger intelligence tied to it.
 * She even alludes to this when she tells Shirou that she'll see him again. Pulling the Devil's soul together was enough to submerge her back into Akasha, but she's not dead. The question, though, is if she'll resist the power of the Tatari's madness.

Saber stays in Kohaku's room as a guard when she sleeps.
We know that Saber went aside to discuss privately something with Kohaku after Kohaku was showing everyone their rooms. In the original Fate Stay Night this was to discuss guarding Shirou while he was sleeping, something he was rather against. Kohaku, unlike Shirou, had no objections; if Saber was comfortable with that; Kohaku had no issues.