Paved and Unpaved Ways? I'm reminded of Frost's famous poem:
Two roads diverged in a wood, and I –
I took the one less traveled by
And that has made all the difference
In other words, "paved and unpaved ways" could refer to a well-worn path and a way less taken, respectively. It's metaphorical.
The paved way is where most people go; it's the safe way. The unpaved way is where the trailblazers and risk-takers go. One might think of the "unpaved way" being taken by inventers, entrepreneurs, and others who take risks and "think outside the box."
Stoney and Daniel are correct in their comments: I'm not providing a standard definition; I'm giving a possible interpretation. You won't be disqualified from this contest because your essay doesn't address the topic in the right way. I'd even guess that a unique interpretation that catches the judges by surprise might count in your favor.
As for the others:
The word meaning has two meanings, so the first one could be interpreted as The Definition of ‘Me’, but I'm guessing most would interpret it as roughly What Defines Me
Because This is What Matters could mean something that is important to you, or something that should be important to all of mankind
The Bravest Place on Earth could refer to any geographic region whose people have shown strong mettle in the face of adversity.