You can have multiple Flashbacks for one character as long as they're not too same-y or uninteresting.
For the melon example, is it really "illuminating" in context? Probably not. That being said, I can imagine a scenario where it would be, because Cowboy Bebop can be really silly sometimes, and that's totally fun to play. Like if a character is obsessed with a melon that only grows on their homeworld, and they haven't been able to find one for 5 years or something. That actually *would* illuminate something about their past, and is totally something that could become something of a running gag.
The Flashbacks don't have to be connected specifically to a Trouble or Bounty. You could use a Flashback to show how you know a particular contact, or why you were kicked out of your gang, or what growing up on Venus was like, pretty much anything. It's basically a way to sneak extra "moments" into a session.
For campaign play, you can use multiple Flashbacks to show how your character's motivation has changed or deepened, explore a new facet of their past, or anything like that.
What about Leverage-style Flashbacks where we learn that the characters had prepared something ahead of time to deal with the current problem or obstacle? (Something like "Oh yeah, I did some research yesterday and set up a fake account that I can use to get past this checkpoint. Let me tell you about it..")