What does "gentleman" mean here? I thought it is only used in formal situation.

What's the "pass" in this situation?

I believe the "pass" here is a letter showing that she is meant to be in that class. It wouldn't be a lavatory pass or a hallway because this is the first day for Bella at a new school. I am unsure what it's like in other countries, but in the United States, students are enrolled in certain classes, and teachers will have a list of their students for each specific class. Bella may need a pass to show her teacher that she is enrolled at this school and that she is supposed to be in this class at this time.

What's "Capitol Hill"?

Capitol Hill is a hill in Washington D.C. on which the United States Capitol (the legislative branch of the American government) is located. It is a vital part of the government and a famous place known to all Americans.

What does the phrase "don't you..." mean?

“Don’t you” is just another way of saying “don’t”! It’s a little more casual in feeling and adds emphasis to the action of advising someone against doing something. Ex: Don’t you think about eating any snacks before dinner. Ex: Don’t you dare think about skipping any more classes.

What does "cozy" mean?

"Cozy" is an adjective that describes something comfortable, warm, and relaxing. Ex: My bed is cozy because it is covered with blankets and pillows. Ex: My new sweater is cozy.

Is it okay to drop "good" when you say "good morning/afternoon/night"? Is it common to say that?

Yes, you can omit “good” to make the greeting more casual! This is best for greeting people you are quite familiar with, or for use in casual situations (i.e. non-work related occasions). Ex: Morning! Did you sleep well? Ex: Night, sleep tight!