Does a Spirit have to pay for the Power Card played via Ethereal Conjunction?

Does a Spirit have to pay for the Power Card played via Ethereal Conjunction?

No, since the card is put directly into play without needing to pay its cost.
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.