When he recommended adding 4 lbs. of stabilizer, he meant buying some cya (cyanuric acid, also called stabilizer) and just adding it directly to your skimmer while the pump is running. It dissolves slowly in your filter, so you'll need to be patient and give it a week before rechecking your stabilizer level or backwashing your filter. The trichlor tabs do contain stabilizer, but since you have a stabilizer reading of 0, it will take awhile before the tabs build it up. I don't use the tabs, instead just add some cya.

Yes to having the pump running when adding the bleach.