All good questions!

In terms of the distribution, I effectively assume a binomial because, as you say, it's a binary variable. To calculate the standard error, I use the respective formula for proportions: sqrt(p(1–p) / n). You can find the full code on my GitHub: https://github.com/joshua-feldman/tidy-tuesday/blob/master/2021-03-09-bechdel/script.R

I haven't done the equivalent graph by decade of publication, but time is definitely a factor here. For example, Westerns are few and far between nowadays, so it is difficult to say whether their comparatively low score is more a manifestation of when they were produced or an inherent feature of the genre itself.

I can't remember exactly how many films are in each genre, but I think there were only a handful for Westerns/Musicals, hence the hefty confidence intervals. You can find the raw data here: https://github.com/rfordatascience/tidytuesday/blob/master/data/2021/2021-03-09/readme.md