Concrete use of convolution algebraic properties
For part 3, there is an old trick dating from the time where computing was expensive. If you have a smoothed version $I_\sigma$ of a picture $I$, you can get a crude yet simple approximation of a  Laplacian from $I -I_\sigma$, illustrated here with the Gaussian:

It is indeed generalized in many unsharp filters, with other weights, or in multirate/multiscale (pyramid) operations.
For part 2, you can convolve twice by Gaussians with a single convolution of a single Gaussian. Yet the most classical (yet rarely understood) is that smoothing first then differentiating is the same as differentiating then smoothing, see for instance Laplacian of Gaussian operator. Many people believe that smoothing first reduces noise. However, one can combine derivatives and smoothing, and save a couple of operations, and even design innovative gradients:

For part 1, this is used a lot in proofs. This is something one uses implicitly very often, for instance in cases when you have an image that you need to convolve a lot of times (scale-space filtering for instance) vs a lot of images you convolve when a filter known a priori.
Related question:

How to choose the size of a Laplacian of Gaussian kernel for filtering images fast?