Peter's attempting to catch a bird is none other then the Looney Tunes character The Road Runner and Peter, with the rocket strapped to his back, is imitating the other famous character Wile E. Coyote.
Peter, Quagmire & Joe are dressed as The Beatles as they appeared on the cover of their 1967 album Sgt. Pepper's Lonely Hearts Club Band. Peter is dressed as George Harrison, Quagmire is dressed as John Lennon & Joe is dressed as Paul McCartney. They wanted Cleveland to be dressed up like Ringo Starr,
The hand in the box Cleveland calls “Thing” is a reference to the character from The Addams Family.
Tim mentioning the “Regal Beagle” is a reference to the bar the characters Jack, Chrissy & Janet frequented often in the 70's/80's sitcom Three's Company.
Robert's references to “Rerun" was a character featured in What's Happening & What's Happening Now played in both shows by Fred Berry.
Cleveland Jr. quotes Carl, the father on the show Family Matters catchphrase "Not Yet!".
Cleveland's “gay architect” comment is referring to actor Robert Reed. His Mike Brady character was an architect on the show, but he was not gay. However, it was revealed after his death in 1992, that Robert was indeed homosexual.
His second comment is referring to the fact that according to the urban legend, Brady mom Carol Brady, played by Florence Henderson and eldest son Greg played by Barry Williams had a sexual relationship while filming. However, according to both of them, Barry simply asked Florence out on one date. Florence declined.