Pepper escaping soap on water
The "soap molecules" are so called amphiphiles: They've got one end that likes water ("polar" or "lipophobic") and the other ("lipophilic" or "fat-loving") doesn't. 

These amphiphile molecules like to form a single-molecule thick film on water, all standing up in an orderly fashion (foot in the water), so if you put a drop of it on a water surface, it spreads very wide, until the film is monomolecular.

Or, until something pushes back. In your case the pepper, when its particles are pushed together so they also form a tightly packed surface. Waiting longer, the soap will likely act as a detergent, wrap the pepper particles in a bilayer, and drag them under the surface.

Search for, e.g., surfactants, detergents for some pictures.

As to your last question: Some water gets dragged along with the surfactant, but no, there is no large lateral movement of water on the surface. The spreading front you see is just the soap.