When House enters the room at the beginning of the episode as the team begins their differential diagnosis, he sets his cane on the table. Dr. Remy "Thirteen" Hadley picks the cane up, (apparently the cane was on top of her paperwork) and sets the bottom end up on a stack of books. As the scene immediately cuts to another angle, the cane is once again flat on the table aside the books, no one having touched it.
When Dr. Kunter is looking at the patient after the diagnosis, he walks up to the glass. However, after the camera cuts back to him from seeing the patient, he has gone back to where he was before moving to the glass.
Foreman and Kutner caught Sophia lying about her parents because her limbic system was active when she was talking about their death, which they said is an area involved in imagination and wouldn't have been active had she been telling the truth. This may be true of a specific part of the limbic system, but the way they said it is false, because the limbic system is responsible for emotions, which would be very active when talking about one's parents dying.
When House enters his office for the first time, Kutner is listing the patient's symptoms and tests. Kurtner states that "Gram stain was negative". Gram stain is a test to identify bacterial breeds in a already cursing infection. Both negative and positive results mean there's a bacterial infection.
It makes no sense for Foreman to ask House for permission to conduct the clinical trial. Foreman was hired by Cuddy to provide oversight over House's team, so he should be answering to Cuddy and be needing Cuddy's permission.