Usually university lectures aren't endless references to studies etc. Not that they shouldn't be rooted in whatever the relevant evidence base or literature says, and that should be stuff the lecturer can point the students to (usually they will have been given that stuff for their reading) but god, what a boring ass lecture that would be. Good lecturers are making a sort of extended argument.
Anyway, the Big 5 personality traits are petty well established in psychology. There are certainly academic disagreements about them or personality tests in gerneral, but it isn't hard to find out how they work, I think there is a Wiki article about that one.
I did it myself some time ago - I scored 60% agreeable, 40% conscientious, 3?% neurotic, 3?% extraverted, 9?% open.
However I do get what he was saying when he was talking about make-up at work. It was in a discussion about 'me too' and he was asking the question about how men and women work together without there being sexual harrassment. He kept saying "it is not entirely clear how men and women can work together" without this existing.
Yeah, actually I think this is a really interesting question and it's unusual to see someone talk about it openly. No one has really figured out how to deal with the fact that there is sexual frisson between men and women when they get together in groups. Obviously this is talking at a society wide level. Sometimes this works out fine (I married my work supervisor which never really caused problems at work or home) but these things do not always go smoothly. Intentions are misunderstood, people get weird, someone gets aggressive, people make decisions based on who is attractive, people stop being able to work together... Banning work relationships is problematic in several ways, from driving them underground to questions around restricting people's right to see who they want.
I think the idea with the lipstick is - maybe if you completely de-gender the workplace, so that anything that tends to emphasise sex differences is disallowed. Which is kind of silly and impossible, but that is maybe the point.