How to solve this problem? Should I manually bevel each edge or there is a another way?
The bevel may be being held back by the “clamp overlap” setting. Turn that off and see what happens.
The stain is normal flaws, and is happening because the shade smooth is averaging the normals between the bevel faces and the large ones that are meant to look flat, affecting the large flat ones as well. The solutions for this problem vary. Most use “Inset Faces” after all modifiers have been applied.