Could someone help me understand what exactly Merry did with the mirror? I understand she showed them visions of the world she's seen, but then they mention something about taking the others' stories and making them their own. Does that mean that while they each looked in the mirror Merry "took" the other people's stories through something like mind reading to verify if they were true or not? 
It seems that Merry's ability improved even more.
As you said, by contacting other people, she can tell whether that person was telling the truth or making up the story as they go. As Merry showed them the mirror to each person, it is assumable that the people got closer to check the fantasy shown in the mirror and touched Merry in the process.
The mechanism of how Merry can distinguish the person is lying or not is not thoroughly explained, but Renko knew about it at this point, and used it to their advantage.
It is perhaps tied to "Merry said that she'd recently started to see multiple worlds at the same time. According to her, all people are seeing slightly different worlds." in the previous album, meaning that she can tell what the other people are seeing. If the person did indeed see this bizarre world, she would be able to peek into their world and distinguish between "complete bullshit" and "actual fantasy".
...by the way, it was hilarious that someone was pretending to be Dr. Latency in this story. It's like ZUN is almost making fun of the people who wanted to be like ZUN, dresses up like ZUN and respond when someone calls out ZUN in the public. This guy is like Dr. Latency's Number One Idiot Fanboy.