Q: Exactly when is a player’s face down card from Dreams And Omens played?

is the face down card automatically played at the start of the appropriate player’s next turn (before that player can draw, play, or use any “on your turn” abilities)?

Hence, if I play Dreams and Omens, the card I give myself is the last of the face down cards to be played. Is this right?

A: It’s just like you said: before doing anything on your turn, i.e. before Drawing, or taking any optional special action derived from a Rule or Keeper.

Yes, your card would be the last of those face-down cards played. Unless the game ends before it gets around to your turn. Mwah-hah-hah-hah….