As House picks up his keys, we can see that they were in the book "Memoirs of Sherlock Holmes" by Arthur Conan Doyle. House himself can be - and is, numerous times - compared to Holmes in many ways.
House says, "We still need Trapper John," which refers to the surgeon of the M*A*S*H in the novel, movie, and two TV series, as played by Elliot Gould, Wayne Rogers, and Pernell Roberts. Additionally, the character appeared in the original series in its first three seasons and then was spun off into his own series, Trapper John M.D., which ran from 1979 to 1983. The reference to Trapper John is to an army surgeon operating under battlefield conditions, and follows a comment that the makeshift hospital room set up in the patient's house is more sanitary than a military field hospital, where the patient was initially treated for his wounds received in action.
Wilson tells House he's wearing a "red badge of idiocy", followed by House talking about the military capabilities of the Confederacy. This is all a reference to Steven Crane's 1895 novel "The Red Badge of Courage", about courage under fire during the Civil War. Wilson is implying that House is a coward for not confronting Cuddy.
When the team talks to the patient through the door, they say, "He must have two locked doors." This system is called a "sally port"--there is a space between two locked doors to control entry and exit.
After Cuddy noticed the cut on House's hand, he jokingly said that he doesn't usually get the stigmata until Easter. Stigmata is when someone spontaneously develops wounds on their hands and/or ankles similar to the injuries Christ endured during His crucifixion.