The rules state that one has to "Finish the game" but I'm not sure exactly what that's asking. Is the game considered finished after the Monoblos, or is it asking that every quest must be completed, making it more of a 100% sort of deal?

you don't have to finish every quest, just the "story" ones.

I'm having trouble understanding what these main quests are. It would be clearer if you said the name of the last mission to be done.

The last mission is the Horn of Monoblos, it unlocks the Hall of Heroes ending credits when you beat it