Yep, bleach is chlorine, Sodium Hypochlorite solution same thing as pool store liquid chlorine but at a different concentration. You can use it all the time, most of us do.

Right, The stabilizer does keep the chlorine from being quickly consumed by the sun.

How much of the cal-hypo (Calcium Hypochlorite - Super Shock and Swim) do you have left? You can use it to chlorinate for a while but it adds calcium to the pool and can cause cloudiness if used excessively.