If an effect happens after a particular step of the Invader Phase (e.g., "after the Build Step") but there are no Invader cards in the corresponding space, does the effect still happen? How about if the card in the space wasn't resolved, or was resolved as a different type (e.g., resolving a card in the Ravage Space as a Build Card)?

If an effect happens after a particular step of the Invader Phase (e.g., "after the Build Step") but there are no Invader cards in the corresponding space, does the effect still happen? How about if the card in the space wasn't resolved, or was resolved as a different type (e.g., resolving a card in the Ravage Space as a Build Card)?

Yes and yes. The steps of the Invader Phase always happen – the Invaders just might not do anything particularly interesting in a given step. For the most common case of this, there is always a Ravage Step on Turn 1, despite no Invader Card being in the Ravage Space.
To cover some of the prominent cases:
  • "Pressure for Fast Profit" (The Tsardom of Russia Level 6) and the "Invested Aristocracy" Event both add extra Invaders on boards where the Ravage Step added no Blight even if there was no chance to add Blight because no Ravage Cards were being resolved. (Fortunately, neither happens on Turn 1 – the latter because it can't resolve on Turn 1, and the former because it explicitly excludes Turn 1.)
  • "Migratory Herders" (The Habsburg Monarchy Livestock Colony Level 1) attempts to gather Towns even if no Build Card was resolved that turn. That being said, if there's no Build Card in the Build Space, this Adversary effect won't actually do anything because there are no matching lands to gather into.
  • "Mining Boom" (Habsburg Mining Expedition Levels 3 & 5) upgrades and potentially builds even if no Build Card was resolved that turn.
  • The "Voracious Consumption," "Ransack and Ruin," and "Visions Out of Time" Events all resolve their "After the Ravage Step" effects even if no Ravage Card was resolved.
  • In The Great River scenario, "Population Spills Across the River" causes extra Build and Explore actions in Coastal lands even if no Invader Cards are resolved in the corresponding Step.