A definite yes for small low-orbit and high-orbit satellites as long as the cube does not rotate. Large moons far away should be fine
The question was answered in the 2012 Physics International paper: "The Gravity Field of a Cube". (Available for free) The authors found stable orbits for a tiny moon or space probe to circle the cube.
The gravity field of a cube is only distorted close to the cube and with a large moon you can presumably have a situation like the Pluto-Charon system where they each orbit a common center of mass, so cubeness shouldn't be a problem in that case. (Though I can't prove it :)
The authors also provide us with great background material for writing a sci-fi story, such as how a lake on a cube will look, and what to do if you colonize the world:
Paper citation:
Consider now a hypothetical cube 12,000×12,000×12,000 km3, approximately the size of the Earth, with the same volume of water and atmosphere as found on the Earth, then we would approximately half fill each face with water and have an atmosphere approximately 100 km thick similar to what is assumed for the atmosphere on the Earth before reaching space. In this case then the corners and the edges of the cube, would be like vast mountain ranges several thousand km high, with their tips extending out into free space. It would therefore be very difficult to cross these mountain ranges and hence we would have six nearly independent habitable zones on each face.
There would presumably be permanent snow on the sides of these vast mountain ranges and people would live around the edges of the oceans on each face in a fairly narrow habitable zone only about 100 km wide as the cube faces rise rapidly through the atmosphere.
Unfortunately climbing the approximately 3000 km high corners does not result in an improved view because the surface is still fiat in any observed direction. However the corners, being in free space, would be very suitable for launching satellites. One would also have approximately sqrt(2) x 6000 km of downhill ski run from each corner, down to the centre of each face.
In order to have a day night cycle we would also need the cube to be rotating. The sun would rise almost instantaneously over the face of a cube however, so that each face would need to be a single time zone and thus the cube as a whole would require four separate time zones, assuming the planet was rotating about the centre of an upper and lower face. The north and south faces in this case would be permanently frozen as they would receive no sunlight except that striking the oceans extending away from the surface of the cube, so there might be a permanent pool of liquid water at the two poles. Launching low orbit satellites around this cube needs special care in order to avoid certain orbital resonances that would create significant variations in the orbit.
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