Right from the off, she's had a storyline that's clearly intended to have her be mistrustful and dismissive of Data before growing to like him. And this is effectively the big change in their relationship. She sets him a mocking challenge and he meets it with near instant success.
No he doesn’t. He actually concedes defeat. Picard has to get involved to resolve the situation.
But, not only does the fact that Data almost immediately solves the "Proper" mystery of the man being strangled with his wife's hair get kind of lost within the overall plot to the point viewers would be forgiven for not noticing it, Pulaski herself never seems to find it out.
That murder was not
the proper mystery he was supposed to solve. It was, as he said, an independent program.
That scene presents a few questions about how Data knew what he knew.
How does he know the victim and the killer spent the day together in a tavern drinking gin? How does he know the killer was motivated only by fear and self-protection? How does he know the killer is the victim’s common law wife? Finally, how does he know that it’s an independent program and not part of the challenge the holodeck created for him to solve?
I’m wondering whether it’s based on an actual Sherlock Holmes mystery. That would answer all these questions.