Consider a standard square sheet lying on the xy plane with edge length n. Is it possible to determine the coordinates (x, y, z) of the point where the vertices of the sheet will meet, when each of the four vertices are pulled upwards and inwards equally, at the same time?

The point will obviously lie right above the center of the sheet. The vertices will be pulled upwards until only a single point (theoretically speaking) of the sheet will be in contact with the xy plane, and then they will be pulled inwards until they meet above the center of the sheet.