As to Carlson, the fact the firing came on the heels of the show you mentioned makes perfect sense--it gave the powers that be an excuse to dump him for a reason other than the fact that he was disliked.  I don't recall the woman's name or job title, but he i the one who makes decisions on who goes and who stays at Fox, and apparently she had a serious and long-lived beef with Carlson.

Honestly however, discussing this sort of thing in detail is not of interest to me.  All I started with was the fact that Conservative networks should hang together because if they don't they will surely hang separately.