Say that you are a bi-dimensional being who lives in the surface of a sphere that expands.

(1) What was here before the Big Bang that set it off?

Well “here” means a point in the sphere and when is relative to that point in the sphere. At the exact time of the big bang (when the sphere surface was a point), I cannot answer. Just after there was a here and a time. Before doesn’t make sense because there wasn’t any sphere surface and no time.

(2) If our universe keeps expanding, what is it expanding into? And what’s on the other side of that?

What’s on the other side of a surface of a sphere? Note that a surface of a sphere is 2 dimensional and doesn’t need a third dimension to expand.

Most people think at the universe as a giant cartesian 3d plane with an absolute time. It is not a good model and brings to paradoxes. Think of the concept of “simultaneous”. The simultaneity of two events depends on the observer because of the limit of light speed. Even cause effect doesn’t apply outside the cone of light.