[icpxseqm] Fairy pawns are the soul of chess

The unique way pawns move and capture in western chess, as compared to shogi and xiangqi, causes pawn structure to be a distinctive and important strategic aspect of the game.  If pawns were to move differently, how would the theory of pawn structure change?

What is the theory of pawn structure in which pawns move and capture exactly one space forward?  One can no longer form diagonal pawn chains.  What about Berolina pawns?

How is the theory affected if each side starts with two (or more) rows of pawns in the front? Possibly of different types.

Consider a piece which can capture any of its 8 immediate neighbors but can only move to the 5 neighboring squares to the front or side.  The is the most powerful piece that seems pawn-like: pawns can't move backwards (but can capture).  It can form interesting pawn structures.