A scene where Peter tells the joke was cut from television. It is however included on the DVD.
The dirty joke told throughout the episode is taken from a joke the character Marty Funkhauser (Bob Einstein aka Super Dave Osborne) told in an episode of Curb Your Enthusiasm (TV Series 2000--).
The 1967 Isaac Asimov short story, "Jokester" dealt with the same topic. Human computer programers look into the ultimate source of all jokes and find that they were secretly developed by space aliens as a means to study the human psyche.
In this episode, Brian, Stewie(and Meg) didn't have any roles, they just waved from a distance before their house while Peter, Lois and the others left them home to start their trip.
The Obama Monument is the Washington Monument painted black. The joke being that it's a big black phallic object.