Intermediate Documentary Filmmaking - S2-E16
Visible crew/equipment: When Jeff runs to the car which his father drives and opens the door of the driver, you can see the camera in the side window of the car. You can see it's a set camera and not a 'handheld type', the one that Abed is using during this episode.
Intermediate Documentary Filmmaking - S2-E16
Deliberate mistake: This episode is meant to be entirely a documentary made by Abed. At the end of the episode, there's a shot where the gang is looking into Pierce's window, except the camera is right in front of the window they're looking through. If Abed was filming this he would literally be right in front of the gang, completely obscuring their view through it.
Answer: The implication is that Jeff will still be teaching at Greendale after Annie and Abed leave. As for who became vice dean? It's never shown, since it's not really important to the series.
TedStixon