Nadeko studies in the same middle school Koyomi did a few years ago, in fact, she transferred there because of him, that's why Tsukihi and her kind off grew appart, they stopped studying together and their old friendship only reignited this past year, when Koyomi started hanging out with Sengoku again. Koyomi was a public school middle schooler, while his sisters always studied in private schools.
And just by chance Ougi is staring at Nadeko's shoe locker, the one where she felt the white snake crawling on her hand for the first time, just a little while after this visit, huh? (both those arcs happen in October)
Araragi's bad memory plays a big part in a lot of the series: forgeting about Nadeko's existence until seeing her climbing down the mountain in Bake; forgeting/ignoring the fact that Kiss-Shot feeds on humans in Kizu; the date of which Black Hanekawa tried to kill him that led to the apocalyptic world in Kabukimonogatari; etc... now his memories of trumatic/important events in his life, like the class meeting and the brainteaser letters, are coming back in full force.
Now for the trivia time!
The desk Oikura was sitting on was Araragi's. She'd been waiting for him, so even if he hadn't had the resolve beforehand to approach her on his own volition, she wanted to clash with him regardless.
Gahara also pretends to pass out after hitting Oikura so her image wouldn't be portraied as the perpretator of the fight to the teachers. She didn't want this to become a reason for the school to back down from her recomendation letter to uni. Her reasoning being "if both of them are down, both would get the sympathy of the teachers".
And about the Monty Hall Problem, if you switch after one of the wrong doors has been revealed, your chances of winning are 67%