Camera calibration and extrinsic parameters for perspective transformation I'm working in a photogrammetry camera and laser application where I need to measure a displacement in a specific plane, this is a quick diagram of the setup:


In order to get proper measurements I'm trying to do a projection from the 3d points to the 2d camera plane  proceeding like this:


First I'm doing the Camera calibration using opencv and a chessboard,
I'm taking a few chessboard shots in different angles and applying the
function initUndistortRectifyMap, I´ll have the distortion coefficients,
intrinsic and extrinsic parameters.
With the intrinsic parameters and the coefficients I'll apply lens correction.
The extrinsic parameters of the previous step are for each chessboard
position and I assume that they won't be valid for my particular
application, so I'm placing a chessboard in the above marked area
as "Measurement Plane" and running again the initUndistortRectifyMap function
which this time it's giving me, I think, correct extrinsic parameters
of the ROI.
Once I've got the "correct" intrinsic and extrinsic parameters I´m
applying the camera matrix to calculate the 2d points (u,v) projection:




My questions are:

Is it the correct way to proceed for this application?

The camera system only reads X,Y pixel points each time, but to perform a 2d projection seems like I'll need the Z, how could I have this last coordinate of the 3d point?
For your application, here are the steps I would follow:


Calibrate the camera and obtain the intrinsics - You have obviously done that and corrected for the distortion already. initUndistortRectifyMap only lets you to precompute the projection. Otherwise, you might as well use projectPoints each time. This is a detail and I would not care about that initially. But it's a good practice anyways.)
The main idea is to find the laser plane (with extrinsic calibration to the camera). To find the extrinsics:

a. Place a checkerboard in front of the camera and leave enough space on the board for the laser line to be projected. 

b. Image the laser line projections together with the checkerboard.

c. Now we can compute two planes with respect to the camera: The plane joining the laser line with the camera center, and the plane that checkerboard lies in (we now the checkerboard). 

d. Intersect the two planes to find the line in 3D space. 

e. Collect all such lines from multiple images and fit a plane to all of them. This is the equation of the laser plane.
To make measurements in 3D, simply shoot lines through the observed laser points in 2D image and intersect the rays with the laser plane. Each intersection will give a single point in 3D space.


This way we also do not explicitly need $(R,t)$. Our extrinsics are rather the equation of the laser plane. 

In case we use a point-laser, the equation also remains the same, as instead of lines, we can collect sets of points and still do the plane fit.

By the way, here is a toolbox and a recent further read.