Continuity mistake: Chang's sunglasses are sitting on the table in front of Professor Hickey and Shirley. When Richie sits down, the sunglasses have disappeared. (00:10:10)
App Development and Condiments - S5-E8
Continuity mistake: Jeff is doing stand-up comedy with a mic stand an a stool with a bottle of water on it behind him. A few shots later, without touching them or anyone moving them, they all completely vanish. (00:12:55)
App Development and Condiments - S5-E8
Continuity mistake: There is a gavel with its handle facing inwards toward the table, facing away from Jeff. When the app developers begin their pitch, the handle of the gavel is now facing towards Jeff. (00:01:00)
Analysis of Cork-Based Networking - S5-E6
Continuity mistake: Abed opens his locker to reveal Britta's spoiler sheet. It rolls open but it doesn't fold open to see what it says exactly, but two shots later it is now open fully. (00:05:30)
Analysis of Cork-Based Networking - S5-E6
Continuity mistake: As Professor Hickey is waiting on the janitor to finish his meal, he looks away towards the hallway. After the janitor speaks, the camera cuts and he is now looking directly at him. (00:03:05)
App Development and Condiments - S5-E8
Continuity mistake: Professor Hickey is rolling up a newspaper in front of him, when the camera cuts away and then back, it is flat against the table. (00:02:00)
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