"Exit" is correct, because the grammar of the context requires a noun. "Exit" is usually a noun, although it can be a verb. "Leave" is usually a verb; when it's a noun, it is used only with certain verbs, particularly "take." The word "dramatic" further suggests "exit," because in the drama, on the stage, in a theatre, when an actor leaves the stage he "exits" or "makes an exit." Stage directions always use the word "exit." So, "made a dramatic..." automatically suggests the word "exit." If the sentences had been "He quickly ______ when his enemies arrived" and "Hannah _______ after fighting with her friend," you could fill either blank with "exited" or "left."

Yes, the word "exit" is the better choice. That word is more commonly used when a noun is called for, but I could also say "take my leave", which is quite formal.

Hi there. You could also say a quick getaway in this context of the sentences you mention. That would be ideal.