When did Bruce Wayne build the Batcave?

Well, the first origin of the Batcave came in 1954’s Detective Comics #205 by Bill Finger, Sheldon Moldoff and Charles Paris, which showed how Batman began to build the Batcave…

Who builds Batman’s stuff?

Lucius Fox helps design most of his arsenal along with the construction of the Batcave beneath Wayne Manor. He, along with Robin, Batgirl, and J’onn J’onzz, play a part in helping Batman defeat the alien robots known as the Joining after one Joining operative poses as Lucius Fox.