So, was rereading and decided to grab quotes that jumped out at me:
As for the rest of these questions:
1. Calcinating Circle: My guess back then was that it was linked to Hikawa's attempt to activate Conception and destroy the world by conducting mass human-sacrifices to demons, hence all the D-events. That the "circle" is the deliberate wide-scale ritual he's doing. In fact, the first thing I thought of was Fullmetal Alchemist and how massacres in certain areas of the country were actually blood-sacrifices to fuel a country-sized transmutation circle.
2. Yukari: One of the next-generation Guardians at Seiyo Academy, but an OC for the quest, not from Shugo Chara. By the QM's description:
Yukari is a third-grader, and one of the grade school Guardians. Her main trick is hydrokinesis. Her chara's focused on surfing. Ami befriended her a while ago, though Yukari tends to see Ami as "this one really cute kid".
3. web-whirlpool-starry-cloud-pattern-thing: Likely the Sea of Souls. Souls in Persona can be connected to each other, expressed in gameplay mechanics by "Social Links" so if Amu could see/feel those, it would probably look like a web. Dia's ability is to navigate an extratemporal dimension in Shugo Chara known as the Road of Stars, which is where Charas come from and allows Amu to access other people's memories too. My bet is the the Road of Stars and the Sea of Souls is the same thing in this quest.
4. Ami: I mean, I've said a lot about what I think about Ami by this point. Though I don't remember it being specified anywhere that Kana can't wiretap other people's mental conversations like Ami. I'd think that she can. The boy she saw in the Dreamlands that had Ikuto-like hair I speculated at the time was probably the Persona 3 protagonist, still guarding the Seal. Especially since Ami was "chasing the moon" at the time - the P3 Protagonist is only there at the Seal, because he's meant to be stopping the moon from falling. So if she chased it, stands to reason she'd find him holding it back.
5. Scavengers: I remember it being said that the only one of the Scavengers who would be able to recognize Smol Miki was Kana, from Amu's memories. So they don't know about Charas. If they have to use Persona Suppressants, they are likely Persona-users. Basically Strega from Persona 3, but with a few more members, a less religious leader and kill using psychic powers instead of shooting people with plain guns during the Dark Hour. "Anti-Penumbrals" were mentioned, so they are likely not just on Persona Suppressants, but also Shadow Suppresants.
My current theory in regards to them is that their Personas are unstable and switch between being Personas and degrading back into Shadow Selves. A further guess is that when Yui Dreamwalks, Shadows are attracted to her like they are to Amu (and likely Ami) during the moments when her Shadow Self is in a degraded non-Persona state, but then attack her during the moments when it stabilizes. So Kana needs to watch her and can usually bail her out when it happens - but my guess is that what may have happened this time was that Yui failed her Overgrowth v Integrity check and attracted the wrong Shadow and then ended up absorbing it, causing her Shadow Self to go completely out of control.
In other words, if Ami is a ticking timebomb, Yui is one that already went off. Ami is arguably more stable, being 50% Alice at least makes her strong enough that it allows her to pick and choose which other Shadows she wants to eat.