Since the deadline is June 22, students have until that date to register (enroll). After that time, however, no one can enroll any longer. (you indicated this correctly by what you said by number 2)
"Enrollment is limited" means that the number of students who can enroll is limited. For example: the enrollment limit could be 100, 200, or even 500 people. There is a maximum amount of students who are able to enroll in the school. After that limit has been reached, then no one can enroll any longer.
Whether the deadline passes or the enrollment limit is reached, whichever comes first, these are the rules set that might prevent some students from enrolling.
To answer your original question, however, whether the word enrollment is countable or not depends on the context. However, I more often hear the singular, uncountable form.
Is this clear? If there anything else you need explained, please let me know.