Here they are:
1. While in Paris, a little girl buys a present for the policeman she has a crush on.
Hint: This movie is connected to Star Wars and My Fair Lady.
The Lavender Hill Mob (unsolved)
Star Alec Guiness is the Star Wars connection. Co-star Stanley Hollaway is the MFL connection.
2. A drunken playboy, a horse trainer, and a doctor are all in love with the same woman.
Dark Victory (solved by Melville)
3. Part of her occupation involves showing people the Grassy Knoll.
Dr. T. and the Women (solved by Tom)
4. The couple meets while sweeping a wooded area for a missing girl.
Hint: During the "present day" of the movie, they are no longer a couple, but both continue to work in the business that gives the movie its name.
Exotica (solved by George)
5. A proposal and rejection is secretly recorded and played back in front of a large group of musicians.
Hint: When they meet, she is singing, he is drunk, and he ruins her favorite lipstick.
A Star is Born (1954) (solved by Roberta)
6. “Stirred, not shaken. That is right, isn’t it?”
You Only Live Twice (solved by Tom)
7. His unfinished home is full of memories of his late wife, so he lets another couple use it.
Hint: This movie is famous for being the last completed film of two big stars who died young, and one of the last for another star who died young.
The Misfits (Eli Wallach lends his home to Clark Gable and Marilyn Monroe; the last completed movie for both; co-star Montgomery Clift made just three more movies before his death.) (solved by Tom)