The Stone of Destiny card
Q1. It says you can swap ANY two pieces. Does this mean you can swap captives as well?
You can swap ANY two pieces, including captives (except you cannot swap a piece on the King space). This includes captives, so the 'classic' way of swapping is to swap one your captives with an opponent's piece on top of a stack.
Q2. It says that 'a player may then pay to activate one or both of his or her pieces that moved'. Does this mean you can switch two of your own pieces and then promote them both? Or does this mean that if you switch with another player both players get to promote?
After swapping, the player whose turn it is can then pay to promote. If you swapped two pieces in a way that resulted in one of your pieces controlling a position, you can promote from that position. If you swapped two of your own pieces, both of which control their positions, then you can promote from both positions. The non-active player never gets to promote.
Q3. Creating 2 Forts: a Blue player uses a Stone of Destiny card to swap the top stone of a RED/BLUE stack with a BLUE/RED stack. This creates 2 forts. Right?
The Stone of Destiny card permits swapping of any 2 pieces, except for those in the king space. So it can be used to swap a single piece from a stack with another piece from another stack. This could create forts. It could, very briefly, create a second fort for one of the two players involved. If a player has two forts in the same region, the owning player must choose between them, so that no more than one fort for each player in the region remains.
Q4. Do you pay the activation cost once for each piece you want to activate or once per card, even if it allows you to lay more than one piece?
An activation cost is paid only once for one card. If the card specifies 2 placement positions, you can activate both for the single payment. In some circumstances this can lead to dramatic 'upward cascading'.