When to use "mark sharp" instead of "edge bevel weight" and vice versa ?
When to use "auto smooth" instead of "shadow smooth" and vice versa ?
The Mark Sharp option is used with either the Edge Split Modifier or when using split normals. It tells Blender that the edge you've marked should be treated like a perfectly sharp edge with no smoothing.
The Edge Bevel Weight is used with the Bevel Modifier. It tells Blender what the relative widths of the edges should be when you use the Weights option in the modifier.
Smooth Shading should be used when you want your entire object to be shaded smooth.
Auto Smooth should be used when you only want sections of you mesh that are already fairly smooth to be shaded smooth, and edges that should look sharp to stay that way. Say you have an object that is mostly smooth but has a few ridges that should be really sharp looking. Using Auto Smooth, you can set edges with a certain angle between them to be kept sharp, while smoothing all the rest.