You can certainly reshape the data to make it fit a 2D network. You could set the width or height to 1 as suggested by DamirTenishev, but you could also set the features/channels to 1 and treat the features as a height, if you wanted to convolve that way (would be a bit strange).
This depends on the engine you use, but in general, yes, of course.
For example, in the TensorFlow height and width are separate variables, so nothing in your way to set one of them to 1 to have 1D data in it.
A convolution layer has a Kernel that is matrix smaller that the image (in many papers 3*3), so this Kernel have learnable values.
The kernel is applied pixel by pixel (and dimensions, for color images), by applied I mean take the kernel centered in the pixel and multiply element-wise with the image and sum all results. This is the output for that pixel, a ...
The proof of the first statement can be found in these Lecture Notes.
Have a look of the Proof of the Claim 1.
Concerning the renormalization trick I do not see easy way to justify this statement. The paper claims:
i.e. adding self-loops to the graph, improves accuracy, and we
demonstrate that this method effectively shrinks the graph spectral