When does Brandenburg-Prussia Level 2's early Stage III card count as Stage III and when does it count as Stage II?
It only counts as Stage II for "What Invader Stage is the game in?" (E.g., for the Second Wave Scenario, the Ward the Shores Scenario, Events, or the rules for playing with Beasts tokens but no Events.) It's still a Stage III card for anything that cares about the Stage of a specific card (e.g., the Escalation rules for Combining Two Adversaries, or Fractured Days' unique power "The Past Returns Again").
It ceases to be special once it leaves the Invader Deck (even if it returns to the deck, e.g., via "The Past Returns Again") or gets shuffled with other Stage III cards (e.g., by the Fractured Days innate power "Visions of a Shifting Future").
If you would Accelerate it out of the deck, instead discard the next Stage II card underneath it (or the top-most Stage III card if there are no Stage II cards left).
Visions of a Shifting Future can't bury it to the bottom of the deck (since it's specially placed during Setup), but can shuffle it to take a chance at delaying it.
However, the Spirit still needs to pay any costs associated with using the Power, whether that's a mandatory "Cost to Use" (e.g., for Absolute Stasis) or a cost associated with a particular choice (e.g., for Hazards Spread Across the Island). You can pick a Power Card with a Cost to Use that you can't afford; in that case, you fail to use the Power and then Forget the card. Tags: Cost To Use, Resolving Powers, Ethereal Conjunction, Absolute Stasis (Fractured Days Split the Sky), 3 more...
It still counts as being resolved "for the first time" (JE rulebook p. 27), since it was turned face-down and returned to the Invader Deck; it has no "memory" of what it did the previous time it left the Invader deck. Tags: Stage II escalation, The Past Returns Again (Fractured Days Split the Sky)
In the Second Wave scenario, where exactly can a power from the first wave be used? What if the power is targeted from a Sacred Site? What about (one of many weird exception cases)?
The power can target any land or Spirit it normally could – no current Spirit needs to have a Sacred Site (or Presence in a land of the correct terrain, etc.) The power is being "played" by the Spirit from the prior game – if it cares about the originating land, or where the Spirit has Presence / Sacred Sites, it can assume that for purposes of targeting and resolution the Spirit has up to 2 Presence in each land. The Power can't affect the playing Spirit (which is not in the game). Technically, Energy/other costs can't be paid, but it's more fun if you say that costs up to 8 Energy / 4 Time, can be paid. (Or rough equivalent, so "Elemental Teachings" can grant 3 of any element(s), "Draw Towards a Consuming Void" could redo its effects once, etc).