Does playing a hidden card at a location start a showdown?

No, playing a hidden card does not start a showdown.

A showdown begins when a battlefield becomes Contested and the turn is in a Neutral Open State (Rule 344). Playing a hidden card from facedown opens a Chain, immediately putting the turn into a Closed State — which blocks a showdown from starting. More importantly, you can only hide cards at battlefields you control (Rule 811.1.b), and when played from hidden, permanents enter at that same battlefield (Rule 811.1.d.1). Since you already control it, no contest over control arises — the hidden card is just entering a location you already own.

Contested status is applied when a unit moves to a battlefield its controller doesn't control (Rule 445). Playing a card is not a Move, so even if the hidden card is a unit, it doesn't apply Contested by itself.