The long pit hall, I found the button to get the chitin helm, when you get to the end of the hall, there is a landing on left with a pressure plate. Throw a rock and it will raise the gate, your in.

How do I get to the book?
Found out how to get book.

can post if someone wishes

I pressume you mean the first step of the trials of thought when you mention the 3x3 room, with portals. its jus about taking steps correctly so you dont get caught with portalls all around you, since when u step on a plate, all the plates connected to it swap from portal to nothing or nothing to portal. so the simple thing is, always move so that you have atleast 1 portal next to you so u have an open path, and as u go along ull get to the switch on the side to open the door and then work ureself to the door.

for the long pit corridor i pressume the 2nd part of time and tide? as my predessesor said there is a button on the 2nd plate to he right to acess a side lane. as for getting threw the gate in the end u need to throw something over the pit at the end onto the pressure plate to get the gate open