Is there a way to attach points along a face like in the capture screen?

I always ask questions with gratitude.

Is there a way to automatically attach a point sketched on the plane to the face by moving it only in the z-axis direction?

The arrow keys allow what is termed “inference locking” during SketchUp operations. The Up key locks (constrains) motion to the blue axis, the Left key locks to the green axis, the Right key locks to the red axis, and the Down key locks to the plane of whatever face the cursor is on when the key is pressed (pressing it multiple times will cycle to a perpendicular direction in some versions of SketchUp). Pressing the same key a second time will release the lock.

For example, this can be used to move any selected entities by clicking (and releasing) once to begin the move, pressing the appropriate arrow key, then completing the move via cursor or entering a distance value and pressing Return.

In your example you have 2 z positions for every point? First thought was ThrowTo plugin…