Audio problem: When Lena and Harold are alone arguing over ice cream bills, Jeopardy is playing on the television. Since the host mentions a $1000 answer, we must be watching the familiar version of Jeopardy, which started in 1984. However, the host does not sound like Alex Trebek, the Final Jeopardy answer is given immediately after a Double Jeopardy answer (without listing the category, or giving time for wagering and/or commercials), and the Final Jeopardy music has been changed.
Other mistake: Trying to impress his new relatives, Rich picks up, and then drops, a piece of rice with his chopsticks. Rich and Waverly act as if the rice has gone inside of his vest, but this would have been impossible since Rich was holding the chopsticks well away from his body. In fact, you can see the piece of rice bounce off of the table and to his left.
Continuity mistake: During Waverly's first chess match, her opponent makes a move, resulting in a board where the white knight and white king occupy non-adjacent squares. In the subsequent close-up shot, the white knight and white king suddenly occupy adjacent squares. Still in the close-up, Waverly picks up the white knight, but then is no longer holding it until partway through the next shot. Finally, Waverly picks the piece back up, places it down, and calls a (rather dubious) checkmate.