Fair warning: when you get there, DO NOT call them drill sergeants. Drill sergeants are usually associated with the Army, and drill instructors with the Marines. So either drill instructors (DI's), or their title (Gunnery Sergeant), but usually both if you're addressing them directly ("Class Drill Instructor, Gunnery Sergeant XXXXX, USMC").
Nicknames usually either come from class team, or candidates will give each other nicknames. Some are a play on their names (like in that video, Tom "Top Gun" Cruz), some because they look like someone famous ("Lex Luthor"), and some because they suck at drill in some way (I'm assuming "Correct Yourself," "Sabotage," and "Extend" fall into that category).