If I move a unit into a battlefield I control, does it start a showdown and give a window for me to play when in showdown effects or for my opponent to play action or reaction speed cards?

No. Moving a unit to a battlefield you already control does not start a showdown and does not open a window for either player to play action/reaction cards (FAQ #8066, FAQ #5090).

A showdown only begins when a battlefield becomes Contested — meaning a unit controlled by a player who doesn't already control that battlefield enters it (Rule 445, Rule 344). Since you already control the destination, no Contest status is applied, and the move resolves without any showdown or response window.

The only exception is if the move itself triggers a card ability (e.g., "when this unit moves") — that ability would create a Chain, allowing reactions during that chain. But no general window opens for the opponent to act.