Very amusing to read that you think the English stereotypically imagine the French to be frogs. In fact it came about as a nickname, not a stereotype, because frogs legs are eaten as a delicacy in France. It is more stereotypical for the English to think of the French as always wearing berets, striped jerseys with a kerchief and riding a bike with a string on onions on the front, or carrying a baguette! Stereotypes are often based on a supposed mode of dress or on stories or myths. Take for example the leprechaun of Ireland. Mythical obviously, but come St Patrick's Day, in the US particularly, little green clad people are the first thing people think of when thinking of Ireland's patron saint. And of course that's absolutely nothing to do with leprechauns.
With regard to the carrots, this is less a myth and much more contemporary. Pilots of WW2 actually were advised to eat plenty of carrots as they are known to contain retinol (vitamin A), which is necessary to prevent night-blindness. This became the overstated myth that 'carrots help you see in the dark'. Which of course isn't true.
So myths generally aren't created. It's truer to say that they arise as a result of the perceptions of people concerning the habits of others. I don't think you'll find many that were created as the intentional lies of one European nature to be used against another. That's not the nature of myths, and it's probably why you can't find many (or any) examples of this.