Darcy’s cap in the vlog is owned by Lizzie. It was probably her dad’s and she dug it out of the attic to represent “Darcy” in her costuming, just as Lizzie got that kid Doctor mirror head band thing to represent Bing Lee.
Does Darcy own a similar hat? Yes. Does Bing Lee wear a doctor mirror? No.
As for everything else… no spoilers.