I feel like they may have intended the trial to be an ongoing subplot through much of the season, and then they changed their minds. Perhaps something else turned into a better storyline, or they realized that a legal focus would be boring? In any case, they wrapped it up in one ep with a rushed cross examination and forced reveal.
I didn't get a clear look at the "evidence" that Laurel showed Moira, but it was on some kind of letterhead. I presume that it wasn't a thank you note from Malcolm to Moira for the evening of hot sex, written on Merlyn Global Group stationary...
Factual evidence to submit in court would be the paternity test showing Thea was Merlyn's child. This would also add some ongoing drama, in that Laurel knows about not just a night of sex, but Thea's true parentage. It means Moira was keeping a deeper, darker secret.
Their father's sordid affairs were apparently well known to the kids so was it worth affecting the court case just because she didn't want her kids to know she had sex 20 years ago? But the paternity of Thea was worth covering up, this is more understandable that Moira would want to keep this secret.
Since Merlyn is officially dead, Thea is actually the heir to his fortune.
I hope the show follows up on these issues and doesn't just sweep them under the rug.