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Stein was an author in the early 1900s. Not really memorable except for her connections with more significant artists like Picasso, Matisse and Hemingway. She happens to fit into a repulsive current agenda.

Fox is a current actress, but I don't see why someone would choose her as an epitome of beauty, unless this is part of the parody.

I finally did think to look up Fox on the IMBD, figuring she was probably an actress.  I never heard of most of the movies and tv shows she was in and never saw any of the movies and tv shows I had heard of.  I'm guessing it was part of the parody since her name is fox, as in "foxy"?

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That one puzzles me too. I think most likely it's just an inside joke with the author, like maybe he had a bad experience with someone named Laura.

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Back in the day, there was 'Lauragate.' A user named Laura, who claimed to be a ɧoɱosɛҳųαƖ with a terminal illness, created an account on FE. Really long story short, it turned out that the whole thing was not only a hoax but that the site admin's live-in-boyfriend (yes), who was the tech-mind behind the FE site, was having a (virtual?) affair with Laura.  As memory serves.  This was about ten years ago.
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Part of the drama was that whoever made the Laura account also made another account (Laura's friend, 'morecoffee,' I believe) and posted that Laura died at one point. Anyways, eventually the whole ball of lies/wax was unfurled and it was pretty ugly. 
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Though it was rather overshadowed by Impygate, I think.
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FE is a really f@#ed up place.  You'd think it'd be enough that the whole place is just really, really vanilla in its traditionalism.  But no, there's plenty of cohabitation, ɧoɱosɛҳųαƖ apologizing, infidelity, fornication, transgender porn, etc. to go around too.

Ah, I wondered whether Laura was somehow connected with Impy.  I don't think I was around for either, and I was banned from FE fairly quickly.

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There's no direct connection (in that the operators of both accounts are totally different people who never met each other, so far as I can tell), but there is definitely a thematic connection in that the sagas involving both parties laid bare the site admin's very twisted tolerance for sɛҳuąƖ impropriety and degeneracy.
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In the case of Lauragate, Vox was the 'victim' insofar as it was her live-in-boyfriend who ditched her-- but still, the fact that she was administering a site designed to draw people to the life, faith, and culture of traditional Catholicism all the while living as she was caused great scandal.  She defended this as well.
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The whole Impy thing was really sick. For those who are unaware, a user named 'Impy' (a man from the east coast) 'came out' as transgender and Vox not only defended Impy but empowered him-- making him an admin, encouraging him to continue with estrogen 'treatments', changing his screen name to 'ClareBrigid', allowing him to post images of the 'transition,' etc. And banning anyone who objected to the moral rectitude of the whole thing.
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The usual refrain from Vox and her supporters was that those who objected were pharisees who just couldn't handle the fact that people have 'pasts.'  A (hopefully obviously) vain retort to the specific incidents under consideration.
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Recounting these things as artifacts of the past makes me feel very old.