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If I have a Micromaster in Bot mode, and I use an action that taps it, such as Inferno Breath or Swap Missions, will tapping it with one of those actions let me activate the Micromaster's Tap ability?

No. Those are mutually exclusive. Tap abilities on Micro Masters are costs for triggering their abilities, so other in-game actions do not allow you to use that ability.
Source: Wizards of the Coast (2019-05-10)    【Link to Ruling】
Tags: Tap