What would be the correct terminology for soft cover on crust? Earths' crust is usually not exposed. Above-water portion is covered with soil, and Underwater portions are covered with various deposits or sediments. Oceans contain Pelagic sediments.(I don't know whether this underwater sediment considered as soil or not).

The soft cover is probably not being considered as Crust. Some sources telling it 
pedosphere (soil-sphere), but with pictures of only terrestrial soil.

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)? If not; then


what is the correct term for the complete soft-layer on Earth? 
Is there any term for part-of skin-of-Earth which is beneath water?
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.