I must concur that it would be called a beanie or a skullcap. I've also heard it called a ski cap.

Unfortunately, I can't help with 2a, since I've only ever gone to schools that based registration times on class year (so that seniors waiting to graduate would get first priority, with incoming freshmen at last priority). I have indeed had to defer things because they were full, but got to take them the next semester when I was bumped further up the registration order.