What would happen if you kept electrolysing non-iodised salt solution, until all the Hydrogen and Chlorine present in it are exhausted ? (Apart from getting poisoned due to Chlorine )

 

Would the remaining Sodium react with the oxygen and form a considerable amount of Sodium Oxide ? Will this form on any of the electrodes or just collect at the bottom of the vessel ?

 

What about other aqueous solutions of ionic salts ?

Does it vary for each salt ?

Electrolysis is rather complex process with many side reactions so there is no single answer. You will surely decompose (or evaporate) all the water. What remains is complex mixture of substances. If you started with NaCl then you will be left with some NaCl, NaOH, NaClO, NaClO3, NaClO4.

 

If electrodes are not completely inert you will also get oxides/hydroxides of electrode material and probably chlorides/chlorates as well. Exact proportions of your products depend on electrode material, temperature, current density and many other things.

Also, once you've decomposed enough water, the remnant heat will cause some water to evaporate and pretty soon there won't be enough water left and current will cease to flow.