Electrolysis of an aqueous solution of sodium chloride

We put some Universal Indicator into salt water and put in graphite electrodes. This is what happened. The solution contains four types of ion: H+, Na+, OH- and Cl-. Opposites attract. The Cl- ions go to the anode on the left and give up an electron each. Two chlorine atoms then bond covalently to form molecules of chlorine gas. This is acidic, hence the red colour and a bleach, hence the white. There was a strong smell of swimming pool in the vicinity! Now those electrons given up by the chloride ions are pumped round the circuit and are given to the H+ ions, which had been attracted to the cathode. Bubbles of hydrogen gas form at the cathode. That leaves the Na+ and OH- ions in solution, Sodium hydroxide is a strong alkali, hence the purple. I have not made much progress in my pursuit of electronegativity as to why some ions stay in solution and some don't. Also, I'd like to know why chlorine bleaches.