When his mom leaves Alan with his pants down, Alan falls trying to hop to the bathroom and goes on himself on the rug. In the next scene that shows Alan at home, he emerges from the room and the runner (rug) is unstained.
In the season 3 premiere, Jake wants the new Final Fantasy. The one he finds in the video store is Final Fantasy 10 (the Greatest Hits version, no less), which was released in 2001, and season 3 premiered in 2005. The most recent Final Fantasy in the show's time line at this point would have been X-2, even 11.
Jake rents Final Fantasy X, but the music (when Jake plays it later on his Playstation2) is the battle music of Final Fantasy II, the original version that was released for PlayStation.