When Bruce and Dick are watching the news reporting that Batman is being sued by The Riddler, the image on the TV is from the same angle in which we see the photographers first come into the scene. How can there be a picture when the cameras are at the other end of the hall?
When an unconscious Robin is being carried from the Batmobile, you can see one brief shot where his cape falls off.
Close up shots of the Riddler tightening the vice on Robin's head show him twisting the knob between thumb and index finger. Long shots from behind Robin's head show the Riddler turning a crank like handle, a totally different attachment to the vice.
Robin drives the Batmobile into a parking lot. In future episodes it is made clear that Robin can't drive the Batmobile as Dick Grayson was still too young to have a driver's license. Dick Grayson would eventually get his driver's license enabling Robin to drive the Batmobile in season three.
As Batman is handcuffing The Riddler, he identifies himself specifically as a duly deputized officer of the law. As a sworn peace officer in discharge of his duties, Batman would have qualified immunity from being sued for assault, slander, and false arrest.
Robin isn't allowed to enter the nightclub because he is underage. However, Robin is a duly deputized officer of the law acting in the course of his official duties and thus would be exempt. Further, Batman, also a duly deputized officer of the law, could bring an underage minor onto the premises if doing so was necessary to investigate a crime.
Bruce Wayne says he is going through his father's old law books. Bruce's father, Thomas Wayne, was a doctor. It would not be unthinkable for a doctor to have law books, as there are a good many pieces of legal knowledge that a doctor would have to be familiar with. The show never explicitly states that Thomas Wayne was not a doctor.
The Riddler tries to steal the Batmobile by using the keyless ignition. However, when Batman is trying to drive off later, the GCPD officers demand Batman hand over his keys, which are standard metal keys. But the keyless ignition is actually the car's anti-theft activator; at about 20 minutes, when he tries to go help Batman in the dance club, Robin is seen folding down a placard with a false label.
As Robin is being carried off after being hit with the tranquilizer, his cape falls off, revealing the appliance attached to his shoulder which holds the dart in place.
The Batmobile accelerates out of the Bat Cave with its turbines screaming. It also casts a shadow to the back of the cave, which comes to a complete stop just as the Batmobile is out of view.
In the opening titles for this episode and the second part Smack in the Middle (1966), the 'POW' sound effects appear to be misplaced. The initial ones before the 'Batman' title appears are correct, but the ones that usually follow the titles (when Batman and Robin punch villains standing behind them, and then on the camera pan across the group of villains) are silent - instead the sound effects are sounded rather clumsily and abruptly over Adam West's credit. This has been corrected on some syndicated repeats, but has been left intact on the recent DVD and Blu Ray releases. This version was also used for the DVD menu background music.
When Riddler bumps into the large vase it tips over. We HEAR it break although we can SEE that it didn't.
While the Riddler may not be guilty of any crime related to the cross, there is still probable cause to believe he is guilty of a number of crimes at the Molaovia event, thus Batman is well within his rights to arrest him.