What exactly happens when $\rm NaCl$ water conducts electricity? Assume a DC power source with $2$ electrodes made of Fe. We dip those $2$ electrodes into table salt water. What happens exactly?


*

*Will $H^+$ and $Na^+$  migrate to the negative electrode by electrical field
or diffusion or a combination of both? 

*Will $H^+$ accepts frist electrons
and then $Na^+$ or both ? But what if we really $amp$ up the current, are we going
to see metal $Na$ at the negative electrode and then $Na$ reacts with
water violently?

*At the positive electrode, should we expect oxygen and chlorine
gas or just the $Fe$ electrode gets eaten away?


Although there are many questions, but I believe there is one general principle that can explain all. Something that can explain the priority of all possible reactions.
 A: To describe the diffusion, migration (under an electric field) and the convection of species we have the Nernst-Planck Equation:
$$ \frac{\partial c}{\partial t} = - \nabla \cdot J \quad | \quad J = -\left[ D \nabla c - u c + \frac{Dze}{k_\mathrm{B} T}c\left(\nabla \phi+\frac{\partial \mathbf A}{\partial t}\right) \right] $$ $$ \iff\frac{\partial c}{\partial t} = \nabla \cdot \left[ D \nabla c - u c + \frac{Dze}{k_\mathrm{B} T}c\left(\nabla \phi+\frac{\partial \mathbf A}{\partial t}\right) \right]$$
To determine which reactions happen during electrolysis we know the Nernst Equation. 
$$ \Delta G=-nFE  $$
Basically a spontaneous reaction occurs when the Gibbs free energy is negative this is useful during redox reactions. For redox reactions the Gibbs free energy is negative and for electrolysis the minimum cell potential required for a reaction is calculated using the Nernst Equation. The thermodynamically favorable reaction can be found using a standard reduction table the compounds that are more positive (higher value) will be reduced and the compounds that are lower value will be oxidised. We note the standard reduction table assumes T= 298.15 K,
an effective concentration of 1M for all species. 
Using a standard reduction table is just a quick way to guess which species will form, if you know the concentrations of the species you can use the Nernst Equation. THERMODYNAMICS IS NOT ALWAYS THE SOLE CONSIDERATION HOWEVER as kinetically some of these reactions are slow and the thermodynamics do not matter.The rate of these reactions compared to one another determines the products produced, the thermodynamics determines which ones can potentially occur. Yet another consideration is whether the products will react with the solvent reversing the reaction. The rate of these reactions can be sped up by increasing the voltage
A: 
...I believe there is one general principle that can explain all. Something that can explain the priority of all possible reactions.

Under normal conditions sodium chloride has a crystalline structure. Each ion from $Na^+$ and so from $Cl^-$ is surrounded by six ions of the opposite charge. From this we learn that one electron from sodium is more on the chlorine side and less on the sodium side.
Since water is a good solvent due to its ionic character it is not surprising hat the aqueous solution of NaCl looks like this:

The attraction between the Na+ and Cl− ions in the solid is so strong that only highly polar solvents like water dissolve NaCl well.
   When dissolved in water, the sodium chloride framework disintegrates as the Na+ and Cl− ions become surrounded by the polar water molecules...
  The sodium and the chloride ions are also strongly solvated, each being surrounded by an average of 6 molecules of water.

And the one electron is still missing sodium, and chlorine with the surrounding water molecules has captured it.
There is a field in chemistry called electrochemistry. It is about the destruction and new formation of chemical bonds by electrical energy. The chemical bonds appear to be strong, but only a few volts are sufficient to destroy or re-form compounds with fluorine.
Using an electric potential difference the ions are moving to the electrodes, of course being in balance in the solution. All other of your questions depend from the material of the electrodes and their electronegativity. 
The electrode could be destroyed (more true for the chlorine side) or the electrode gets plated (more true for the sodium side). All this is really a question for CSE.
