Shareholders say Fox's disregard for the truth opened it to the $787 million settlement it paid out to Dominion Voting Systems and other such claims.
Todd, who has hosted the program since 2014, said it's been an "honor" as he handed the reins to his successor.
The Fox News host's sexist attempt at levity fell flat given the channel's history of sexual harassment.
The fired host was interviewing someone on his internet show when he slipped in the misogynistic dig at his old network.
The prime-time host inflated the importance of the conservative network to an extreme, critics on Twitter said.
Instead of photos, the front page featured text messages sent by UNC students as the shooting was taking place, such as “Are you safe?” and “Where are you?”
Thompson will take over from an interim team of CNN veterans who led the network following the departure of Chris Licht.
The Fox News host also appeared to borrow a line from "Seinfeld" before praising the former president's looks.