Corrected entry: When Bridget Fonda meets and moves in with her boyfriend, she not only gives him no "back story" - meaning a lie - about her life, but when he asks her questions, she fumbles and makes things up on the spot. She was trained at an elite top secret government agency - there is no way they would not have trained her in how to lie and give herself a good cover story.
Corrected entry: When Bridget Fonda wakes up in a room, the guy that's in charge shows her pictures of her staged funeral and tells her Plot 48, Row 12. Later in the scene when she asks him what happens to her if she refuses to become the spy or whatever she becomes he says Row 48, Plot 12.
Correction: Well, Plot 48, Row 12 is already "occupied". Row 48, Plot 12 might be an ironic reference to another grave.
Correction: This is NOT a film mistake. It is your opinion of her actions and how you think she should have acted. It is your rewriting of the screenplay. Is it impossible for a woman befuddled by love to act in a slightly ditzy manner? No. It is unlikely a trained spy would, but unlikely is not impossible, and therefore this is not a film mistake.