Other mistake: Probably throughout the show - but definitely in Season 1 EP 2 - the picture of Central Park West (?) as it sits shows the middle left turn lane arrow pointing the wrong direction. The arrow shows a turn that would cut across two lanes of traffic that you were traveling with to make the turn. It shows the turn arrow pointing across the oncoming traffic.
Continuity mistake: When Emma and Zuri are watching the horror doll movie, Jessie comes in for a little bit and you can see that the popcorn bucket is completely full, but after Jessie leaves and they see the part with the eye, Zuri puts an empty popcorn bucket on her head. Wasn't there popcorn in it a few seconds ago?
Character mistake: In the beginning of the pilot episode, Jessie told the cab driver that her father is in the Marine Corps. However, it is shown in this episode that her father is actually a lieutenant colonel in the Army. It's highly unlikely that Jessie wouldn't know what branch of the military her father was in considering how much she spoke about her military upbringing throughout the series.