That is very true.

As for the bit about Mulgrew, no. She did lend her narration to a 'documentary' that promoted Geocentrism but so did Lawrence Krauss, renowned physicist. Both regretted it. Kate said on Facebook:

I understand there has been some controversy about my participation in a documentary called THE PRINCIPLE. Let me assure everyone that I completely agree with the eminent physicist Lawrence Krauss, who was himself misrepresented in the film, and who has written a succinct rebuttal in SLATE. I am not a geocentrist, nor am I in any way a proponent of geocentrism. More importantly, I do not subscribe to anything Robert Sungenis has written regarding science and history and, had I known of his involvement, would most certainly have avoided this documentary. I was a voice for hire, and a misinformed one, at that. I apologize for any confusion that my voice on this trailer may have caused.

This is the Slate article she's referencing.

No she is not.

Star Trek's Kate Mulgrew Says She Was Duped on Film Narration

She narrated a geocentrism documentary (or contributed voice over for it?) but she said she was misled into doing it. So, no.

But that doesn’t excuse Harry Kim of being an incel!

The only info I could find on Google says that she was duped into doing narration for the film The Principle, that apparently supports some shitty pseudoscience about the sun orbiting the earth.

Nothing about being a flat earther, that I could find.

Dan Olson has a pretty good video about that one time some geocentrics tricked a bunch of physicists and Kate Mulgrew into being in their lame documentary. https://m.youtube.com/watch?v=icwDF8wRgF4