The Universe is proposed to be flat. It’s size is 13.7 billion light years. (From where? Is the Universe measured from Earth? Or is it measured 6.85 billion light years from Earth on both sides?
Also,does this mean that the Universe has a top and bottom because it’s flat? What if my ‘Starship’ went straight ‘Up’?
I realize there is no up and down in space, but how cannot there be an ‘up’ (or down) if It’s flat?

Also, in my mind I can imagine traveling 15 or even 20 billion light years away. Does that mean that something exists out there because I imagine it does?

“Flat” means that (on the large scale, at least) it lacks curvature, not that it’s two-dimensional. Another way of saying this is that if you drew a very large triangle in the Universe, the sum of its angles would be 180 degrees. By contrast, consider the surface of the Earth, which is not flat: You can draw a very large triangle on the surface of the Earth such that the sum of its angles is greater than 180 degrees. For instance, a triangle formed from the equator, the Prime Meridian, and the 90 degree longitude line would have each of its angles be 90 degrees, for a total of 270 degrees.

One can envision a universe where all large triangles have a sum of angles greater than 90 degrees: Such a universe is said to be spherically curved (like the Earth). One can envision a universe where all large triangles have a sum of angles less than 90 degrees: Such a universe is said to be hyperbolically curved. And one can envision a universe where all triangles, no matter how large, have a sum of angles of exactly 180 degrees: Such a universe is said to be flat, and our Universe appears to be an example of this.

The universe is much larger than 13.7 billion ly. It’s age is 13.7 B years, but radius of the observable universe (meaning, the observable sphere with us in the center) is larger than that because of the expansion of space. See khanacademy’s video for an explanation

Ignoring the premise here for a moment, I would say that you can’t imagine travelling 15, 20, or even 1 billion light years away. You can imagine a number and a word but you have no concept of the actual distance involved, not in the same way you have a concept of the distance from your house to the nearest shop. (That’s not a comment on you, I don’t think anyone can truly fathom large numbers in the way you are suggesting.)