After a card with the Omega keyword is played, the current step of the game ends. The active player cannot play, use, or discard any more cards for the remainder of the step. Any pending effects and triggers complete their resolution, then play continues to the next step.
Yes, Autocannon will destroy Duskwitch, and yes, the Omega keyword will still apply. Playing Duskwitch causes Omega to trigger, and that ability will finish resolving even if the creature is destroyed immediately upon entering play.