Why won't Evee open the selected material in compositing I remember that in older versions of Blender the current material would be selected, if you chose an entity with a material.
As you can see in compositing editor, there should be some type of diffuse and image node.
You are mixing up some things here:

The node-based material definition process is not the same thing as "compositing". They share the feature of being node-based but they are not the same thing. 

Since they are two separate things, Blender 2.8 split the editor into two: the "Compositing" editor and the "Shader Editor" (Eevee doesn't really matter here, this is an unrelated change).



The one that you find in the "Compositing" workspace is the compositing editor. The one you need is in the "Shading" workspace. 



It is also in whatever workspace you want, if you select the editor type to be "Shader Editor".