How does destiny work in the Lunaverse? Also, will the concept of destiny play any role in future Lunaverse stories?

Vaguely, and it's definitely possible to fight fate. I am a fan of prophecies, but they always take the form of something to the effect of, "If X, then Y", with various factions trying to either cause or prevent X. Using the show as an example,

"On the longest day of the thousandth year, the stars will aid in her escape."

So in this case, "the stars will aid" would be the X, and "her escape" would be the Y. Theoretically then it was possible to prevent Nightmare Moon from escaping the moon by preventing the stars from aiding her.

Nowadays I'm kinda hating destiny/chosen one tropes.