At the moment, I can't answer the part about the shape of the core. I can answer the part about the magnetic field.
The Earth rotates, and because of that its shape is that of an oblate spheroid, so any body that rotates must take this shape. Now, you can object, saying that the body in question is not originally a sphere but a cube, and so it will not become an oblate spheroid. However, there will be some elongation along the equatorial axis, and so the planet will become a rectangular prism.
Furthermore, a rotating object must have approximate axial symmetry - in other words, if you look down on it from the axis of rotation, it should appear the same shape in all directions. So now we've shown that the planet must be at least a cylinder. However, we already said that the object will bulge out along the equator, and so it will become an oblate spheroid.
You could object to this, saying that the planet doesn't have to rotate. Ah, but it does, if you want a substantial magnetic field like the one you're thinking about. The Earth's magnetic field (and that of other planets) can be thought of using dynamo theory. As Wikipedia says,
In the case of the Earth, the magnetic field is induced and constantly maintained by the convection of liquid iron in the outer core. A requirement for the induction of field is a rotating fluid. Rotation in the outer core is supplied by the Coriolis effect caused by the rotation of the Earth. The Coriolis force tends to organize fluid motions and electric currents into columns (also see Taylor columns) aligned with the rotation axis.
Therefore, you need a rotating planet to have a magnetic field, and a cube-shaped planet could not have a magnetic field.
I am, admittedly, a bit late to this interesting series of spherical-planets-questions (I wish I hadn't missed most of it!). Samuel had a nice explanation of the reasoning for a cube-shaped rotating planet:
The cube is sustained 'magically'. It was shown previously that a cubic world would be impossible otherwise. That was what my stated assumption was addressing.
See, this is what I get for coming in late. :-) It seems the original idea (after some searching) was of jamesqf, in a nice answer here. The point was a bit of a buzz-kill, and to keep the whole scenario magic of some sort had to be invoked.
I'll stick with my guns on this one, especially because of the use of the reality-check tag. I'm much more reluctant than most people to use magic - which is why many of my answers are a wee bit disappointing - but I do believe that a given situation should use as much science as possible, especially with this tag choice. But if magic is allowed, then feel free to invalidate this answer! I'll be disappointed, but I'll understand. I suppose life needs a bit of magic to it, though I prefer the metaphorical kind.