Without more information, it's impossible to say who Lauren is. She could be an ordinary girl, a character in a made - up story, or someone known within a particular group.
Alexis might confront Beckett directly. She could say something like 'I think you're cheating on me. What's going on?' This is a very straightforward approach.
I don't know who Alexis is in this story without reading it. But it's likely that Alexis is either the main character or an important figure in the events that unfold in the 'creepy subway' setting.
A lawyer could be an important character. This lawyer takes on Alexis's case, believing in their innocence. The lawyer has to battle a corrupt legal system or powerful opponents who want to see Alexis locked up for good. The fanfic could show the lawyer's strategies and how they build a case in Alexis's favor.
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Perhaps there have been some rumors or whispers in their fictional world that have reached Alexis' ears. Even if they are baseless, they can plant the seed of doubt in Alexis' mind, making her think that Kate is cheating.
She could be defensive of Beckett. Since Alexis probably has a good relationship with Beckett, she may try to convince Castle that he's wrong and that Beckett didn't mean any harm. For example, she might tell Castle about all the positive things Beckett has done for her.
In the 'father does know best the lauren chapin story', Lauren Chapin is probably the person whose life and experiences are being told in relation to her father's wisdom. She could be an ordinary person whose story of family dynamics, with her father being the source of great knowledge, is being shared. Without more details, it's difficult to say more about her, but she's clearly an important part of this narrative.
It could be that Bella has a love interest that Lauren also fancies. In fanfiction, love triangles are common. If the object of affection shows more interest in Bella, it would make Lauren jealous. For example, if there's a guy both of them like, and he always chooses Bella over Lauren.