Batman assumes that the Green Hornet is a criminal, despite having met him before and knowing him to be a crimefighter.
In the beginning when the Green Hornet and Kato enter the "Stamping Room Keep Out" door, from inside the next room where Colonel Gumm and his henchmen are, the lettering and peep hole on the door has disappeared. After the short fight when the Green Hornet and Kato leave the room, the lettering and peephole are back.
At the start of the show Bruce Wayne holds a supposedly rare stamp with his fingers. Any true collector would handle it only with a pair of tweezers to avoid damage.
When Batman and Robin are stuck on the glue pad on the wall, they act like they can't move their arms. But it is their capes, not their their arms, that are stuck to the pad, meaning they should be able to swing their arms freely.