Why go through all the trouble when they could just make you quit?

Perhaps you could build a case for constructive dismissal if you document everything including the sexism email

They could just not respond at all to the email. They could say the person receiving it didn't understand what the email was asking so deleted it. They could say that the managers never made any such statements and thought the employee (who they would also say had other performance or HR issues) was trying to entrap them, and that they fired the employee for the alleged entrapment.

Document and report. Don't try to catch someone unless you're advised to do so by a lawyer or law enforcement or the like.

Oh sure, I implied to sue only if a response was received