Yes to both of your questions.
When you are on the manage enrollments page, do you see a cohort enrollment option there? Is the eyeball slashed? If so, then the enrollment method is paused and won't update based on cohort enrollment at the site level.
I'd suggest heading to Moodle's YouTube page - Mary has made fantastic videos regarding many Moodle features and they are helpful to watch - scroll through to find one on cohorts or enrolling users and there should be a video about this