Bitoku

Yes — once a die is unlocked you place it on an empty Forest space and may immediately perform the Forest action and/or one Building action (follow the die-value rules). (pages 6, 9)

Yes; you may perform one Forest Action (only if your die’s value is equal or higher than all other dice in that region) and/or one Building Action (your die must meet the Building’s minimum value), but you cannot do more than one of each per die placement. (page 9)

When you cross you move the die to an empty Hill space and pay an Act of Contrition by reducing the die by 1 (or by 3 if it was showing 6); the die’s value does not restrict which Favor you may take, and you cannot cross with a die showing 1. (pages 10–11)

Yes — if you use a Building Action on a Building owned by another player, that Building’s owner automatically receives the Property Bonus shown on the tile (from the common supply). (page 9)

Turn order is set by the dice in the Home of the Great Spirit: the player whose die occupies the highest position goes first, then the next highest, etc.; if a player has multiple dice only their highest one counts, and if no dice are present the turn order stays the same. (page 12)

Yes — you may permanently discard a Pilgrim from your player board to use a locked die immediately, and that Pilgrim is removed from the game. (pages 6, 9)