Does the pawn's initial double square move have a special name? Does the pawn's initial double square move have a special name, akin to how special capturing move that the pawn has is called en passant?
No, it's a standard move. By definition, pawns are pieces which can move 1 square forward or 2 on the first move, and can capture diagonally. These actions don't have special names as they're easy to define, but en passant requires a bit more explanation (as most pawns never do it in any particular game).