The scene in which Gary Oldman's character Stansfield talks about his appreciation of Beethoven to Mathilda's father was completely improvised. The scene was filmed several times, with Oldman giving a different improvised story on each take.
Natalie Portman was originally turned down by Todd Thaler (the casting director) due to being too young, but she returned to the auditions and performed the scene where Mathilda laments the loss of her brother. Luc Besson was so impressed with the depth of emotion she summoned during the audition that he gave her the role.
Mathilda wants Stansfield dead in order to avenge the death of her brother, which was the only member of her family that she cared about. Although Stansfield kills the rest of Mathilda's family, her brother is actually killed by Stansfield's henchman Willi (you can hear Malky scolding him for it when Mathilda walks by the entrance to the apartment, right after the raid). This means that she is actually present to witness the death of her brother's killer, as she is in the room when Léon shoots him.
Léon's milk rises and falls in his glass, depending on the camera angle, immediately prior to the charades game. He pours a less than full glass for himself when his back is to the camera, but it is at the top line when the camera faces him. As he talks with Mathilda, there are several shots back and forth with the level changing back and forth each time.
It's Always Fair Weather (1955), directed by Stanley Donen and Gene Kelly, and starring Gene Kelly.
Why does Leon save Mathilda's life?
As part of his code of conduct, Leon's main rule is "no women, no kids," and Mathilda fits both categories. He knew that she would suffer the same fate as her family at the hands of Stansfield and his men if he didn't open the door. Although he would not be the one who pulled the trigger, he would have felt responsible for her death, since he could have prevented it.
What are the differences between the theatrical cut, the director's cut, and the extended version?