Robin is quite happy to jump into the air on the cover of Batman 4 (Winter 1940).
The Joker returns in a very good tale by Finger, Lane and Robinson. It sets the pattern for many of his Golden Age tales, opening by showing how he survived his apparent death in his last appearance, and of course, appearing to die again at the end of the story.
In between death, he spends this story running a circus troupe, who perform at private parties in wealthy peoples homes. They case the places, and then return later to rob their hosts. Batman and Robin wind up in a fight with some acrobatic thieves, and noticing the pattern of the thefts, realizes that the circus performers are the criminal gang.
Robin gets a good page of action before getting captured by the circus villains.
Batman tracks the bad guys to the Joker’s lair, in a castle rigged with things intended to freak Batman out, and kill him. But he avoids the traps, frees Robin, and the Joker appears to die – but comes back in the next issue.
Gamblers going after a football player make up the plot of this Finger, Kane and Robinson story, but the story is not one of the really important things in the tale.
Right as the story begins, Batman gets into an altercation with the bad guys, and shoots at them. There is an editorial note that he doesn’t carry a gun, or kill with it, but he clearly has less problem using one than he will in later days.
One of the hoods believes that Batman must be Bruce Wayne, using the connection both of them have with Commissioner Gordon as his reasoning. To prove it, some of the gang follow Batman and Robin, while other encircle Wayne Manor.
This story establishes a passageway between an abandoned barn far from the house, and the house itself. There is no element of this being the Batcave, but time would expand this into it.
Outwitted, the gang continue with their plot, kidnapping the football player, who Batman impersonates in the game, and later frees.