Earl says that Catalina spent four days on a raft coming to America. In all subsequent telling of how she came to America she came in a box.
A casino would never accept a set of car keys as currency for a bet at a table. Joy's mom could not legally gamble the car away because she was not the legal owner. Randy was the legal owner. He could have returned to the casino the next day and demanded the return of the car.
When Catalina wins the car in the radio station contest, the DJ says she has to fill out paperwork first and asks for her social security number. When she runs away because she has no SSN, the DJ hands the car keys to Randy and walks away. It's not explained why Catalina has to fill out paperwork yet Randy doesn't.