Sorry, but how is anything of what you've said an example of "putting women 'in their place'"? Do men not have their credentials questioned or their opinion dismissed online, or is it misogyny purely because it's men doing it to women?
As I already said, it happens to women disproportionately. If you'd like to learn more about when exactly it's misogyny, again, read Manne's book, the heart of which is precisely defining the term.
I'd like to know how they arrived at the conclusion that it happens disproportionately.
I personally arrived at my conclusion through, as I explained, long observation of social media. If you'd like to understand the contents of a book, well, there's a common solution for that.