Obviously, ABGD is a square with CG=2√2.
CF is perpendicular to x axis, CE is perpendicular to y axis, GH is perpendicular to CE,
Therefore, CG=t-√2, CH=CF=- 1+t/(√2), OG= 1,
So C(t/(√2), 1-t/√2).
Actually, I don't quite agree with the establishment of the department. Too much trouble.
For example, the first question:
Connect AC, let AC∩BD=F, let EF∨PC, EF∩PA=E, obviously, PC∨surface EBD,
AE/EP=AF/FC=AD/BC=(√2)/t? , you don't need to ask for the coordinates of point C.