What would be the correct terminology for soft cover on crust?
what is the correct term for the complete soft-layer on Earth?


Sediments.


  Is there any term for part-of skin-of-Earth which is beneath water?


Marine sediments.


  The soft cover is probably not being considered as Crust


It is definitely part of the crust. When discussing mantle-crust issues, then crust is usually in the context of crystalline rocks, but it depends on context.


  But it is not clear to me : exact how-much portion is being called pedosphere? Is it the whole skin of Earth (including underwater-depositions)?


Pedosphere refers to terrestrial soil and sediments, and it usually discussed in context with other "spheres" such as atmosphere and lithosphere. Sediments (and marine sediments) is a neutral term.

As always, in Earth Sciences, there are several terms for the same thing. There is no "correct" term, just "better" when discussing said thing in various circles.