On the blackboard behind House is the phrase "Tesla Was Robbed." Nikola Tesla was once offered $50,000 to create improvements for the Edison Company, and after the work was finished, he was not paid. Here there are two tie-ins, one is that the patient offered House $50,000 for her treatment, and the second is a joke that most of the applicants for House's team are there working for nothing since chances are they'll be fired.
The intro is a tribute to the movie 2001: A Space Odyssey (1968), replicating Bowman's trip through the stargate.
The doctor that's fired for guessing Neville Chamberlain (#23) is the ER doctor from Alone (2007) that House told to send him her resume.
House tells Dr. Jeffrey 'Big Love' Cole to get drunk as part of a test for the patient, but because Dr. Cole is a Mormon and is not allowed to drink due to religious beliefs, he refuses. House then asks the doctor if he would pull an ass out of a pit on the Sabbath. This is a reference to Luke 14:5 of the Bible, where Jesus implies that breaking a rule, such as performing no work on the Sabbath or imbibing alcohol, is not absolute.
Wilson says, "Since she's not a dead cat, it's scientifically impossible for her to be in two places at once," and House replies, "Physics joke, don't hear enough of those!" This is a reference to the paradoxical thought experiment known as Schrödinger's cat.