The 30 second video shot by Howard "Physicists Gone Wild!" is a real video which is still available in YouTube.
This was the first episode written after the 2007-2008 Writers Guild of America strike. During the hiatus enforced by the strike, CBS officially renewed the series for a second season.
As Penny (Kaley Cuoco) and Sheldon (Jim Parsons) are walking up the stairs, and they discuss Sheldon's disagreement with Leonard (Johnny Galecki), Penny asks Sheldon "How do you feel?". Sheldon looks a bit confused and states "I do not understand the question". This is in homage to Star Trek IV: The Voyage Home (1986), the computer used to retrain Spock's mind asks him the exact same question ("How do you feel"), which Spock likewise responds "I do not understand the question".
Sheldon says that Occam's Razor suggests that someone threw the letter in the trash can. Occam's Razor is the principle developed by the philosopher William of Occam in the fourteenth century that states that of many competing explanations for a phenomenon, the simplest one is likely to be the correct one.
The swelling music Leonard, Sheldon, Howard and Raj are pantomiming to when Penny walks in, commonly referred to as the "Theme from '2001: A Space Odyssey,'" is the introduction to the 1896 tone poem "Thus Spoke Zarathustra," by Richard Strauss (1864 - 1949), inspired by the novel of the same name by Friedrich Nietzsche (1844 - 1900). The introduction is also known as "Hymn to the Sun," from the beginning of "Zarathustra's Prologue". However, the guys' pantomiming indicates they're referencing 2001.