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….