Throughout the series, it is explicitly shown and spoken that The Study is off limits to anyone but Bruce, Dick and Alfred. However, Alfred has no issues with Barbara's entry into The Study.
During the highlight reel of Batman & Robin's adventures, the scene of the Batmobile driving down a dirt road while mines explode around it is identified as being from one of King Tut's capers. It is actually from Catwoman's scheme in Better Luck Next Time (1966).
During the episode, there is a scene where Alfred has to hide the Batphone at Wayne Manor from Barbara Gordon, with Barbara looking around at the decorations and furniture and Alfred hiding the phone under a dust rag. However, though she never looks directly at it, the uncovered phone is in Barbara's sight twice--once just before she enters the study, and once just after it starts beeping--but she doesn't notice it at all.