A: No, you can’t use the Central Axis card to Exchange Keepers with another player. It may count as yours during your turn, but not to the extent of being able to trade away what is essentially public property.
It would be like, if, every Wednesday, the cow on the commons got to be yours, and you could milk it. You could even enter it into a contest and win! But you can’t decide to trade it away for a new refrigerator just for yourself.