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.
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.
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.
As the patient lies in the hyperbaric chamber, the defibrillator is used, which starts a small fire. However, the procedure had not yet started; since they were still prepping the patient and the door was wide open, the oxygen level in the room would have been normal and there would have been no problem using the defibrillator.