Suppose I have the following image (i.e. I have the coordinates of all points in 2d so I can regenerate lines and check where they cross each other)
Now suppose I have another image of what I know to be the same lines:
How can I determine plane rotation and Z depth on second image (asuming first one's center was in point (0,0,0) with no rotation)?
What you see in the second image is a projection of the first, after a rotation. So treat it exactly like that. Meaning, you have 4 lines in $\mathbb{R}^3$. You know their equations. Furthermore, you have another 4 lines in $\mathbb{R}^2$ corresponding to the second image.
To solve, parameterize the family of possible 4 lines in $\mathbb{R}^3$ that project onto the 4 lines in $\mathbb{R}^2$. Find the matrix that takes the original 4 lines to generic quadruple in the mentioned family. Now write equations to ensure that matrix is actually a rotation. You should have enough information for there to be at most one quadruple that can actually be gotten from the original lines. If not, then your question has multiple answers.
This is a common computational problem in computer vision. Here are two sources (among many).
(1) Aaron Bobick, Georgia Tech: "Two arbitrary views of the same scene." PDF download of lecture slide deck.
CS 4495 Computer Vision: Homographies.
Another source:
(2) Jana Kosecka, George Mason Univ.: Uncalibrated Two-View Geometry (PDF download).
