Under US law, all evidence that the prosecution discovers must be given to the defense before trial; this is known as "discovery." Thus, "Adam's trump card" should have been turned over to Jean Loring, the defense attorney, prior to the trial going to court.
Under the US justice system (and most justice systems based on English law), cross examination may only consist of topics brought up during direct examination. This means that Laurel could only bring up "Adam's trump card" if Jean Loring, the defense attorney, questioned Moira about it herself. The only exception to this would be if Moira was called as a prosecution witness, and thus examined directly by Laurel, but this is highly unlikely to happen in Moira's own trial.
The Kairyu class sub mentioned as being beached on this island could not have transported anything to the island during WW2 as it was a short range midget sub. Subsequent shots of the sub's interior show it vastly larger than a midget submarine.
Given Laurel's prior relationships with both Moira and Oliver, she should never have been the second chair for the DA in Moira's trial, much less bumped to being the lead attorney when the ADA fell ill. This goes beyond "saying no to an assignment in her first month" and becomes a serious conflict of interest and possibly even an ethical violation. It's understandable that the writers needed the dramatic conflict, but, in real life, it could easily lead to a mistrial.