So, to reply to the some of the questions about advancing through the school, I think I somewhat remember how to. The goal here is to open the door to the Sports Equipment Room to make something that'll open sliding doors. After that, you need to make it to the door in the Principal's Office, unlock it, and advance to the next area.
First: Head to the Nurse's Office, fight the girl on the bed and defeat one of her ribbons. This initiates the next event I believe.
Second: The ribbon ends up getting ripped, so it can't be opened to use the sliding doors. Afterwards, you have to get the key to the Sports Equipment Room. To do this, you have to survive a trip to the Changing Room via a girl's loafers. I think these loafers were red, and in one of the shoe lockers at the front of the school.
Third: Once you get to the changing room, I'd advise you find the Warp Point up in the vent, so that you can come back if you die. Reason I say this is because you have to hop into a bunch of hampers full of different students clothes to find the key. I think this happens during afternoon class, and so you only have a few tries before they find you, and eventually kill you. You can also just prevent this by just waiting through the day cycle again. There's text that let's you know that the students might come back.
Fourth: Once you find the key, you can take it to the Sports Equipment Room by the track. There's a ton of stuff in there, but you're looking for a specific item that can be used to open the sliding doors.
Fifth: After you're done with this, you can open most of the sliding doors around the school. The one you need to get by, which is in the Principal's Office, requires another key though. This key is on the Principal's Desk, but if you try to get it, you'll either fight her, and likely lose, or just get a game over scene. To get this key, you have to fight the Student Body upstairs. Depending on other, optional events you've done around the school, there may be either one, two, or all three members actually there.
Sixth: After winning this fight, there'll be a cutscene where you're taken by the President of the Student Body to the Science Lab. There ends up being a huge explosion that gets the attention of the Principal. At this point, she's too preoccupied with this incident to come back to her office all the time. That allows you to get the key, opening the sliding door in her room. This will get you to another room where you can uncover the entrance of the side hole to the outside of the school, as well as travel down to the next area in the game.
Bonus One: Given how many optional events happen in this area, I understand why its so easy to get lost and sidetracked. To start, you can go into the Student Body Vice President's shoe locker, which is the one with the sneakers. Upon her defeat, I think you'll be able to fight the student's doing P.E. outside the school. Defeat them too, and the Student Body Vice President won't show up in the fight against the rest of the Student Body.
Bonus Two: At night, the Health Committee President will be in the downstairs bathroom, and you can fight her. Upon her defeat, she won't appear in Student Body fight either. This also leads to the bathroom she was in being closed. This allows you to head to room 3-B, pick up a key normally inaccessible, and unlock the library. Trying to get this key before fighting the Health Committee President will result in a game over scene. In the library, you'll have access to cards with student info on them. Usually this would help with identifying who has the key to the Sports Equipment Room, but due to the fact that most of us are either playing with really scuffed translations, or just no translation at all, I don't think it matters.
I think that should be everything big that happens in the school right now. There are of course the shoe locker fights with other students, and trying to fight the Principal at all, but those require some serious grinding. They're best saved until you've explored the other areas in the game.