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If I use Swap Missions to untap a tapper character and then tap an untapped character, and that same untapped character has a tap ability - does Swap Missions trigger that tapped ability?

No, tap abilities (ones with the tap symbol) happen when the player activates them on their turn and dont happen if the character is tapped in any other way.
Source: Wizards of the Coast (2019-07-29)    【Link to Ruling】
Tags: Tap