The song that Winston (Lamorne Morris) sings at the end of the episode is the same song that Nick (Jake Johnson) was singing in the shower. "Groove is in the Heart" by Deee-Lite.
This episode takes place two months after the season one finale.
School budgets are typically done by the year, not on a day to day basis and it is unrealistic that Jess would simply be let go and leave the very same day during a school year.