Magnets are a wonderful and often misunderstood element of nature and this is an excellent question. The answer, in brief, is that there are electrons in the middle of a magnet, and that there is magnetism there too, we just don't notice it there. And you are perfectly correct, the magnetism there cancels out, and I'll try to explain how.

So firstly, when we are talking about a magnetic field, or an energy field of any sort really, we define it based on what we can observe. No one can see a magnetic field, but what we can observe is the effect of the field on a particle or on other sorts of things. We know that magnetic fields exert forces on moving electrons for example. Now, magnets always come with two ends, we call it a dipole (di for two), a north and a south. Theoretically, there isn't a very good reason for them to always come in sets of two like that, and actually, physicists are looking very hard for magnetic monopoles, ie: magnets with just the north end, or just a south end. If they discover those, it will be an exciting day in physics, but that's another story.

Getting back to ordinary dipole magnets, we say that there is a force on the ends because if we bring a metal object close to those ends then it will feel a force. The force it feels is actually a combination of two forces, the forces exerted by the north end of the magnet and the forces exerted by the south end. When you are closer to one end than the other, then the closer pole will exert more force and the forces from the other end can be ignored. When you are in the middle of the magnet, the metal object is being pulled more or less equally in both directions, and so the result is that the object doesn't really feel a force in either direction, they cancel each other out. So that's why it seems as if there is no magnetism there, but there is still a magnetic field. In fact, you can verify this by placing a compass near the side of a magnet, the compass will line up with the magnetic field and point north.

This is a very interesting question! Here is an experiment you can try if you have the right materials: Take a bar magnet and break it into two pieces, and now try using the broken ends to grab something metallic (like a paper clip). You should find that the new ends should behave just like a regular magnet now! This is because the entire bar of the magnet before was magnetized, but only in a particular direction. Generally, the magnet has been configured such that the ends of the magnet are doing the attraction/repulsion work (imagine lines of magnetism coming in and out of those ends). This means that near the middle of the magnet, the direction of magnetism is parallel to the surface (the lines of magnetism run along the sides of the magnet instead of coming out of the magnet). This is why something like a paper clip does not stick to the sides of a magnet but rather seem to be attracted towards either end.

The magnetic field is strongest at the poles of the magnet, but weaker in the middle of the magnet and is not nonexistent. The magnetic flow goes out of the north end and then back into the south end so the field in the middle actually points towards the north, it just is significantly weaker.

If you put a magnet onto a surface and sprinkle some iron fillings onto it and then gently disturb the fillings so that it can align itself, you can see that although most of it will cling to the opposite poles, some will be quite close to the middle of the magnet.

The magnetic fields are equally strong in all directions, which means that, yes, they cancel each-other out.

Gravity does the same thing: there is no gravity at the very center of the Earth, because you would be being pulled on equally from all directions.