Having core sexist beliefs isn't a prerequisite for making a sexist comment.
Exactly, many people think "I'm not a sexist" and then say lots of things, presuming then that everything they say cannot be sexist. I take care to say "sexist talk/action". Tends to get people's backs up less.
You can do "s/sexism/racism/g" for all of the above as well.
Exactly. I've made this same post several times, only doing a bare minimum of s/race/sex/g