This wasn't really a TIL but it's also RStudio specific. But there are a few examples of some helpful snippets you could include in RStudio to make your life easier - I'm too lazy to list them out so check out the blog posting https://jozefhajnala.gitlab.io/r/r906-rstudio-snippets/
I think C would be the typical answer if not for the word "approximate". That alone makes me think they want them to use 2 instead of the slightly more exact 1.96 when constructing the interval and in using 2 that does lead to (D).
I think you're confused on the rules of thumb. The binomial can be used in any case where it's appropriate. The normal approximation is typically assumed to be 'good enough' if p*n > 5 and (1-p)*n > 5. That is what you were checking so the normal distribution approximation to the binomial...
I didn't look too much into it but if your data is arranged such that observations in the same group are in consecutive rows then serial correlation being detected makes sense since consecutive rows share a common random effect.