If the earth was somehow cut in half can you fall of the edge of the one half of the earth that you're in?

No. Gravity would no longer be acting on you perpendicular to the earth's surface due to the Earth being only half a sphere, assuming the other half somehow vanished. As you approached the edge you would be pulled at a greater angle back towards the center of mass making the approach to the edge seem like an ever increasing upward slope, until you got over it and it would seem to be a downward slope, where you could presumably tumble until you reached the center. Joy.