I think the question you have asked here really touches on a peculiar mystery of physics. As I'm sure you already know, there are both positive charges and negative electric charges, as well as positive and negative parts of a magnet. The weird thing - the thing that nobody understands - is that the positive and negative parts of a magnet ALWAYS come paired together while positive and negative electric charges can be separated. This is so mysterious that many physicists have said that it must not be true, and folks have been searching for a good long while now for these magnetic "monopoles" - that is, a magnet charge without its partner.

Now, it is useful to think about positive and negative electric charges first. Suppose you made the same sphere with positive charges on the outside and negative charges on the inside. First of all, say you specify that the number of both charges are always the same. In that case, if you add up all of the forces from all of the charges, you find out that the total force on any other charges outside of the sphere is: ZERO! On the other hand, if you have a tiny bit more positive charges than negative, the whole sphere acts like there is a positive charges at the center. A little extra negative charge and you have something that acts like it has a negative charge at the center (as long as we agree to stay outside the sphere).

Because of the weird property that there are no magnetic charges (in other words, positive and negative poles of a magnet always come in pairs) you find out if you make a sphere out of magnets and you add up all of the forces from all of the magnets, they should add up to zero. There is no force and what you've made has ceased to be a magnet at all. If there were a force, it would look like a tiny positive or negative magnetic charge without its partner. As I've said, that appears to be impossible so it must be that you always get zero.

Of course, what I've said applies only to spheres. If you make things in other shapes out of charges, either electric charges or magnetic charges, the forces don't exactly cancel out to zero anymore. However, if you carefully measure the forces from a set of magnets, you should never see an effect similar to that of single magnetic charge.

Like I said, it's a piece of physics that is still not even remotely understood.

Good question!

If you mean something that is truly spherically symmetric, which means it is the same no matter which way you turn it, then the answer is no.The reason is a law of physics known as the magnetic Gauss's law.

I'm not sure what you might know, Brent, about the laws of electromagnetism, so I'll try to explain Gauss's law for electricity. If you have a spherical electric charge, you get a spherical electric field. Gauss's law says that, if you draw an imaginary sphere around the charge, the product of the area of the sphere and the value of the electric field on the sphere is directly proportional to the charge.

The magnetic Gauss's law is exactly the same, with one exception. Even though many physicists believe that magnetic charges exist, there isn't any solid evidence that anyone has ever seen a magnetic charge. So this means that there is no such thing as a purely spherical magnetic field. In the spherical magnetic Brent mentions, there would be a sphereical magnetic field between the core and the surface, so you can't make a magnet like that.

This isn't to say you can't make something that is very close to a spherical magnet; it would just have to have a cut or hole in it somewhere.

This is a genuinely interesting question... The answer is that such a magnet cannot be made. Effectively, such a magnet would be a source or sink of magnetic field. Currently, all known magnetic fields have circular field lines, with no endpoints or start-points. A particle with such a field does have a name, however, it is called a 'magnetic monopole'. In theory, such a particle could exist, however, several searches over the last 40 years have not been successful. (Monopoles would be very useful both theoretically and for engineering if they could be found).
However, all known sources of static magnetic fields(all particles carrying a magnetic moment) have (source and sink free)circular fields which cannot be superposed to make a composite source or sink.

Yes, theoretically. I do not see any conceptual reason why not. There is one reason why it is may be somewhat impractical which I will describe below.
For now though, try on the following thought experiment and see if it makes sense to you:

I assume that you are familiar with standard linear magnets. Now imagine taking a sphere and slicing it up into a large number of wedges like you would slice up a pizza but in 3 dimensions. So you are left with a bunch of sharp wedges which are sharp on the end which belongs in the center and kind of flat on the side which belongs to the outer surface. Another way of looking at this is to imagine slicing up the earth into sections by cutting down along the longitude and latitude lines straight toward the Earth's center. Now take these wedges and magnetize them just like you would atypical linear magnet (suppose you make the sharp end "north" and the flat end "south"). If you reassemble the wedges, you will have what you were asking about.

Here comes the practical difficulty: since opposite poles repel each other,as you try to bring the magnets together, they will resist. Even if you make the magnets relatively weak, the resistance force grows fast and will strongly resist letting you put the magnetized pieces back together.Nonetheless, I can imagine you being able to force them back together and hold them in place with glue or some kind of outer encasing. I suppose now the question remains - what would you do with it?

AMENDMENT:

Upon reflection of the other responses, I would like to make the following amendment to my response.
My amended answer is "no, not really" and what I wrote is thought-provoking maybe, but fundamentally wrong unless I clarify the following details:

You certainly CAN take a bunch of linearly magnetized slivers and place them in a spherically symmetric pattern with all of the "north" poles facing inwards. BUT - the problem comes in when you try to bring them together into a sphere (here is where I ran into trouble before).
As you begin to tighten the magnet wedges together the circulating field lines will start to be compressed into the gaps between the magnets. As you close off these gaps,you increase the density of these lines which represents an increasing force(repulsive force in this case) between the magnets. It is fundamentally impossible to close the gaps to zero. Thus you cannot make a magnetic monopole (not even an apparent one). But you can make a device that is somewhat close to what you ask for if you use enough energy to get those repelling magnets together. Further hampering your efforts though will be the demagnetization effects which will take place if you try to get the magnets too close together. As you subject the tips to the individual magnet wedges to greater and greater fields from the other magnets, you will eventually overcome their material properties which hold the tips "north".
In other words, if you did use the supreme force required to crush the magnets together, you would achieve a demagnetization of the spherical body.
Again, you are left without a magnetic monopole. Finally, if you did decide to chop off the tips of the wedges and leave just enough gap to allow the magnets to stay magnetized, you would see a substantial drop in the strength of the "south" magnet surface due to cancellation effects. The magnet sphere's surface would end up only looking like a "south" pole in the center of the wedges (it would only be a very local effect). I suspect that this result is not what you had in mind.