Yes, it's okay to use trichlor tabs in the skimmer--but keep your pump running 24/7 while doing so. If you turn the pump off, you'll have a skimmer full of high-chlorine, low pH water that will hit your pipes and pump on startup, and it's not good for them. Yes, go ahead and add the polyquat at this point, then get your filter tested as soon as you can, per Ben's instructions. Otherwise, it sounds like you've got a handle on it--it's just a matter now of adding a good dose of POPP--pool owner patience and persistence!! (And the skimmer sock will never hurt..... )

Janet