The Excelsior Acquisition - S3-E16
Character mistake: When Howard finds the stack of paychecks in Sheldon's drawer and wonders why Sheldon hasn't cashed them, Sheldon explains that he's saving them to purchase items that have yet to be invented and he doesn't trust banks. But just keeping them in his drawer doesn't do anything, because checks are voided after a set period of time and they would eventually become worthless.
The Staircase Implementation - S3-E22
Character mistake: In 1-14 "The Nerdvana Annihilation", when Howard says they should put the time machine into the elevator, Leonard states the elevator had been broken for two years. But in this episode, we found out the elevator broke seven years earlier. (00:03:10)
The Gothowitz Deviation - S3-E3
Character mistake: Sheldon refers to using the spray bottle as negative reinforcement when it is in fact positive punishment. Reinforcement involves encouraging a behaviour by either adding or removing a stimulus whereas punishment involves discouraging a behaviour by adding or removing a stimulus. Sheldon would know the difference.
The Adhesive Duck Deficiency - S3-E8
Character mistake: When Sheldon is placing his order near the beginning of the episode, he states "In the mid-18th century, King Rama IV of Siam..." King Rama IV was born 18th October 1804 and died 1st October 1868. His reign was during the mid-19th century (1851-1868).
The Einstein Approximation - S3-E14
Character mistake: When working in the Cheesecake Factory, Sheldon says "Bonne appétite," with a hard T at the end. He should know that this is incorrect and would be the first to remind others that the correct phrase is "Bon appétit" (pronounced "appéti" - it is masculine and the t at the end is not pronounced).
The Einstein Approximation - S3-E14
Character mistake: Sheldon is an extreme germophobe to the extent he doesn't even shake hands. There is no way he would be climbing around in a ball pit.
Suggested correction: If he was acting normally (for Sheldon), he wouldn't. However, Sheldon isn't acting as he normally would due to lack of sleep, and his obsession with solving a problem.
The severity of his phobia would not be minimized by lack of sleep.
Sheldon's mind is distracted by the problem he is trying to solve. He grabs other people's food, he allows Raj to touch his food, he handles dirty cutlery at the cheesecake factory, All characteristics that would be present if he was acting normally. The fact that he is in the ball pit is simply an extension.
Severe sleep deprivation could certainly affect how Sheldon reacts to phobias. The brain requires sleep to function and with a severe lack of sleep, the brain is no longer able to send the signals that alert Sheldon to his fear of germs.
The Vengeance Formulation - S3-E9
Character mistake: When Leonard, Sheldon, and Raj are watching Kripke entering his office with the board of directors, Kripke says "President Siebert", yet Siebert is not among the people standing with him.