Maybe you are looking at it the wrong way square!
No, Magrathians.
Right!! 
Notice from Pasta’s picture that the oceans would be a bulge on the cube.
How high would the oceans be? 10 Ft? 100Ft? 1000Ft?
If there were no oceans, would water run uphill in the center of one side?
Define “uphill”. The most sensible definition of “downhill” is “the direction in which water runs”, so using that definition, no. You could also define “down” as “closer to the center of the planet”, but by that definition, water on Earth[sup]3[/sup] would still run “downhill” (and also by that definition, we must conclude that the Mississippi River runs uphill!). If, however, you defined “down” as being closer to the flat surface of the cube (a rather impractical definition, if you ask me), then yes, in that sense, water could run uphill on our hypothetical planet.
Hmm… true. The angle to each would be significantly lower than any real Earth mountain you could observe. But I think once you started to move on the surface of a face, you’d definately feel the direction of gravity change.