I thought I'd chime in on this. Stretch absolutely has something to do with the quality of the cheese. Grande has the best stretch on the market, best flow and highest temperature bake-off qualities before browning. Polly-o comes in second and after that, mileage varies. If you want to get the most stretch from your cheese, don't freeze it and and don't shred it into tiny shreds. The bigger the shred or dice, the less likely the cheese will burn and it will increase its melting qualities. Another trick to getting great stretch and less burn off is to put the cheese down first and dab the sauce over (very little sauce like for a Napoletana or a Lombardi or Patsy style pie) I slice my cheese and put down the slices, which gives me a really creamy cheese melt, that doesn't burn in the wood-burning oven.
The temperature of the cheese has little to do with its ability to stretch, however it is preferable to work with cold cheese. Freezing mozzarella messes up its melting capacity--owing to the loss of moisture during the freezing process. Whole milk mozzarella stretches and melts better than part skim--but that varies per brand too Grande's part skim melts better than most of the whole milk mozzarellas on the market.
I've conducted zillions of these tests over the years, so I've had plenty of good and bad experiences