Most certainly you shouldn’t tell your bosses anything about this person. It’s entirely her business what and if she wants to reveal to her superiors

I see no reason to treat her any differently. You could make choices how much time you want to spend with her outside of work but at work it shouldn’t really matter what mental or physical illnesses she has. I see no reason to be afraid of her.

It’s entirely possible she knows about FMLA. I am not sure why would she need to mention it to you? You mean discussing with you or her bosses? That’s not something people even talk about, not because it’s a secret but because it’s not that interesting to talk about.

I don’t know if she made it up and why. I don’t see any benefit in her telling people she had psychotic break if she really didn’t. What would be the point?