How Is Laplacian of Gaussian (LoG) Implemented as Four 1D Convolutions?
I think Chris Luengo's answer is perfect.
The trick is that you can calculate the 2nd derivative of the image (Using Finite Differences -> Convolution) and then blur it with Gaussian Filter.
Since both are seperable kernels you can do that by 4 1D convolutions.

If the input image is given by I


Calculate Ixx (The 2nd derivative on x direction) using convolution.
Filter Ixx with 1D Gaussian Kernel along the x direction.


Do the above for the y direction as well.
The LoG image is the sum of both.

To answer your questions:


See my explanation above. I think it matches your understanding.
This is the very definition (The Discrete) of the LoG.
I don't see the contradiction between #62 and #63.