Are there ever any moments during your opponent's turn where you can play an action spell, other than a showdown with no chain?

No, there are no moments during your opponent's turn where you can play an [Action] spell outside of a Showdown.

Your ability to play [Action] spells is strictly constrained by the state of the game and the nature of the [Action] keyword.

The Breakdown

1. [Action] spells outside of Showdowns: According to Rule 154, a spell can only be played during an Open State on its controller's turn, provided there is no Showdown in progress. When it is your opponent's turn, you are not in your own turn's Open State, so you cannot play an [Action] spell.

2. [Action] spells during Showdowns: The [Action] keyword is defined as a permissive keyword that grants permission to play a card during Showdowns on any player's turn (Rule 806.1.c.1). As confirmed by FAQs 4805, 6605, and 6353, this permission is limited to Showdown Open states—meaning you must have focus and the chain must be empty.

3. Why you cannot play [Action] spells in a "Closed State": If a chain exists (a "Closed State"), Rule 309.1.a states that only cards and abilities with the [Reaction] keyword can be played or activated. Because an [Action] spell lacks the [Reaction] keyword, it cannot be added to an existing chain.

Summary of Constraints

Addressing Triggered Abilities

It is a common point of confusion to wonder if a card effect that allows you to "play" a card from your hand bypasses these timing restrictions. According to Rule 358.3, when you are in the process of playing a card, the game performs a legality check to ensure the card has the appropriate permissions for the current game state. Since an [Action] spell generally lacks permission to be played during a Closed State or on an opponent's turn outside of a Showdown, these restrictions typically apply.

Note: I could not fully verify the interaction of triggered abilities bypassing timing restrictions against the current rulebook, as specific card effects may provide exceptions.

Conclusion: Unless a card explicitly grants you a specific timing exemption (e.g., "You may play this spell as if it had [Reaction]"), you are strictly limited to playing [Action] spells on your own turn (Neutral Open) or during a Showdown when you have focus and the chain is empty.