The Theban Tribunal of 1200

In which Xavi and Constantine must meet their gauntlet and choose their mage names. Xavi must argue for the conviction of Megehia on charges of interfering with Mundanes and consorting with infernal forces, and where Constantine must find a covenant that will accept him as a member which is not located in Thessalonika (his adoptive parens haaaaabeen growing rather bleak over the past few years and Constantine gets the impression that he considers Aretemis and himself to be something of a disappointment)
Also all apprentices (including those being gauntleted) have an opportunity to be interviewed as to whether they have any complaints to lodge against their parens...

Is finding a covenant Constantine's actual gauntlet?

yes, your current parens has described it as a "challenge to your social abilities".

Did he have any advance warning of that? Would he have had an opportunity to prepare/freedom to choose last couple of seasons?

The last season before the gauntlet he would have known.

How early does Constantine arrive?

[The Theban Tribunal (event) structure is at least 4 days as there are 4 phases and each starts on a new day. That said, each phase seems likely to take more than one day excepting, perhaps, the last which seems a very prescribed ritual where the old leaders leave their posts and the new leaders are sworn in. This also ignores that magi from the Pyle “hosting” the tribunal arrive much earlier to prepare the island and certain other “pregame” events like the apprentice interviews and whatnot. Anyway, seems like it would be safe to say the tribunal itself usually lasts a week.]

Xene arrives on Delos with her parens and is quite noticeably stressed for her upcoming gauntlet. She has had little appetite and been sleeping poorly.

Constantine arrives with a chip on his shoulder. He is fed up with House Jerbiton, but careful not to be open about his feelings. He has a few paltry belongings, and the book he wrote. In fact, he'd like to be gauntleted under any house other than Jerbiton, even if that's unlikely, he intends to make an effort. The first thing he does is try to figure out who is attending tribunal from Hedyosmos, as he's heard it's a subterranean, Terram focused covenant, and that interests him.

Ausculator of Guernicus, Aiakia of Guernicus, Rtsiphon of Tytalus, and Celaenea of Bonisagus are all present at the tribunal as members of Hedyosmos. You have arrived the day before the official beginning of the tribunal with the announcement of decrees (the trial will be on day 3)

[Rolled 17 on OoH Lore, how much does he know about them?]

Ausculator: you know pretty much everything in the book.
Aikia: the youngest of Ausculator's apprentices, still follows him as part of the mystery cult, not much else
Tisiphon: A quaesitor bordering on vigilantism, though he carefully follows procedures in declaring wizard war
Celaenea: recently gauntleted seeker, known to be a young hothead with delusions of grandeur and probably crackpot ideas she is researching

Constantine will go to see Tisiphon.

Ioannes will early arrive together with Red Cap on a ship and a Red Cap hand him over to the young magus responsible for the Kindergarten.
(Might be even one of the player who is responsible for the Kindergarten this tribunal)

Ioannes is 11, and up for apprenticeship, not kindergarden…

Tisiphon raises an eyebrow at the apprentice, just within visual range of his parens. "Apprentice." the word serves as both greeting and question.

(Sorry I thought the Kindergarten was for all children and teen waiting to become a apprentice)
Edit: Ups did react to a talk not meant for me so removed it.

Not sure where you got that term- Kindergarden is german in root (child garden) and, at least locally, refers to the academic class for 5 year olds.

Constantine bows his head slightly. "Quaesitor. May I meet with you at your convenience?"

From my knowledge are the Kindergarden as we know them a quite modern invention. That and because of the large girl in the picture was why I assumed the "hermetic kindergarten" (TSE 33) was to keep all the children and teen who are supposed to become apprentice under control.

that is the name of the story seed, not the actual name of the organizing condition
Tisiphon waves to a nearby alcove. "I have time. I can extend my parma for you, if you would like."