If I play a spell outside of a showdown, does that start a new showdown?

No — playing a spell outside a showdown just opens a Chain; it does not create a Showdown. A spell on an empty board puts the turn into Neutral Closed, and the only thing that happens is FEPR resolving the chain (Rule 344 vs. Rule 310.2).

Showdowns only begin when control of a battlefield becomes Contested during a Cleanup while the turn is in a Neutral Open state (Rule 344). The indirect exception: if your spell moves or plays a unit to a battlefield in a way that applies Contested, that will stage a Showdown which opens in the next Cleanup — but it's the Contested status doing it, not the act of playing the spell.