When Sheldon responds to Howard that the authorities might not approve of bringing a woman from a honky-tonk to a secure government facility, he is clearly expressing a sarcastic response. It has been well established that Sheldon does not have any sense of sarcasm, so this style comment would be completely outside his character.
When Leonard and Stephanie make out on the couch, Leonard's napkin disappears between shots and Stephanie's changes position. Although enough time passes between shots to rearrange the napkins, given the circumstances it's unlikely they would have taken the time.
In the scene where Howard, Rajnesh, and Sheldon are in the Mars Rover Control Room, the clock behind Howard can be shown to be jumping back and forth between different times, sometimes jumping backwards in time in later scenes.
When Raj is typing on the computer trying to solve the problem of the Mars Rover, he clearly is expecting an instantaneous response from it. Even at its closest to Earth (54.6 million kilometers), signals from Earth to Mars would take more than three minutes to get there, and the return signal would take another three minutes to arrive. As an astrophysicist, Raj must know this.
When in the Mars Rover Lab, Sheldon is faking surprise that they don't allow strange women to use $200million NASA projects, but the initial build and launch cost of the Mars Rover Project was $820million.