Well, it's not impossible to make 'down' perpendicular to each side of the planet. Everything with mass creates a gravitic pull on everything else. You are drawing Earth toward yourself, but the planet has so much more mass that when you jump you create a gravitic pull on it so minuscule in comparison to its pull on you that its unnoticeable. And some parts of your body are more dense than others, which parts create more spacetime distortion than the less dense parts. In this cube world, if you had a gradual increase of density of mass toward the edges and corners just right, it would do exactly what you envisioned.
Imagine an ocean covering one of the corners. That would be a bumpy boat ride. I'm not sure that the moon would make water rush around dramatically, at least not in comparison to our world's oceans, with the same moon at the same distance. If the planet's density increased at the corners, those corners, while closer to the moon at times, would still have more mass fighting for the lead position in gravity. Sure, there would be an effect, but I think it would average out to much like oceans on Earth.
If you threw a ball over the edge of the world, the forward momentum would push the ball effectively 'up' from the perspective of the new current side, stealing much of the 'forward' momentum, shifting of gravity would cause it to arc, and the ball would take a steep curve down to the ground.