Do All Great Stories Begin (or End) With Tragedy?

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Tragedy? No. A motivator to seek out a goal? Yes. Tragedy just happens to be an easy motivator.

Pretty much this. It's not so much that tragedy is the best way or even the only way to get the ball rolling, but it is one of the most reliable ways to get the story moving. It provides the main character with the motivation to accept the call to adventure or at least break up the status quo so that they can't reject it. Since Star Wars has been in the spotlight these days, Luke is a good example of the latter:

[spoiler]He initially planned on remaining home in spite of his desire to leave the planet, but after his uncle and aunt were murdered and his home destroyed, there was nothing for him to go back to.[/spoiler]

But yeah, tragedy isn't necessary for beginning a great story. A classic example would be Tolkien's The Hobbit:

[spoiler]Bilbo doesn't leave his comfortable life in Hobbiton because of tragedy. In fact, beyond curiosity and an adventurous spirit (or sympathy for the dwarves, as was the case in the movie), he doesn't really have a personal motive for going off on his adventure.[/spoiler]

On a side note, Harry Potter is kind of a mixed example:

[spoiler]While the death of the Potters is undoubtedly a tragedy, the event also marks the first victory over Voldemort and an end of a long period of conflict. Compared to the years of violence and terror that preceded it, the story's time period is relatively peaceful...at least until the end of Goblet of Fire.[/spoiler]