Why don't electric eels shock themselves?

When a living being is shocked or stunned in such a way, it is because the electric current produced by the eel disrupts and overrides that creatures natural electric current (nervous system). Because the electric eel is generating the current themselves, the organs responsible for producing this Bioelectrogenesis naturally communicate with the nervous systems and the brain, allowing it's own nervous system to cooperate with the electricity generated.

So while an electric eel cannot shock themselves, if they snuck up on another electric eel, they could probably shock and stun it depending on the voltage produced.

Would it be possible for a sixth sense similar to the Electric Eel or Shark's ability to sense bioelectric fields to function outside an aquatic environment?

 

Realistically, probably not, or very weak. Salt water is a much more efficient conductor than air due in a large part to the sodium (which coincidentally is the exact element used in the electric eels to produce their electricity). However, this can be combined with Magnetic bones that have been found in the human nose, specifically the sphenoidal/ethmoid sinuses. They exhibit magnetoception in humans, so clearly the human brain is capable. Magnetoception in humans has also been achieved by magnetic implants and by non-permanently attached artificial sensory "organs". So I could see these Electric Eel type organs having such a knock on effect.

Could electromagnets be reliably weaponized in the form of armor, such as a bracer?  What I mean by this, is could you design a solenoid to fit inside a piece of armor or weapon that could be activated in various ways to either attract or repel other ferromagnetic materials?  Could said solenoid also function as a small, but functional gauss cannon?

The electric organ is made of electrocytes, lined up so that the current flows through them and produces an electrical charge. In order to produce a coilgun, these lines of electrolytes would need to be wound around a metalic barrel of some sort to draw through the ferromagnetic material. An electromagnet functions the same way, in fact the magnetic field created by an electromagnet is proportional to both the number of turns in the winding, and the current in the wire. Since the current can adjusted, the strength likely can be too.

How would someone actually kill via electrocution if that were the aim?  Would it require multiple contact points, or could a circut be formed using a single contact point such as a hand placed flat against the other person?

Death can occur from any shock that carries enough current. Small currents (70–700 mA) usually trigger fibrillation in the heart which is reversible via defibrillator, but large currents (> 1 A) cause permanent damage via burns, and cellular damage. The heart is most devastated by foreign electricity, next is the brain. One contact point should be enough if the victim is grounded. Two would be required if airborne, or the victim is properly insulated. I could see such a technique being used for torture. Also, it's likely that the executioner would receive burns and cellular damage themselves as a cost (which works wonderfully from a story standpoint), and these would take time to heal.

How do magnetic materials actually function, and what determines which side attracts and which repel?

For that I would suggest checking out Magnets on Wikipedia: http://en.wikipedia.org/wiki/Magnet
Should be loads of material there.

What is the difference between static and current electricity, and would it be possible to achieve "miniature lighting" affects from organs similar to those on the Electric Eel?

Again, wikipedia is your friend: http://en.wikipedia.org/wiki/Static_electricity

Hope that helps