As cheese melts, it goes from white, creamy and stringy (I've heard Neapolitans call it 'filante') to bubbly and oily. Taken further, it will start to brown. The larger the piece of mozzarella, the longer it takes to melt/lose it's stringiness and reach the bubbly/browning stage. So, if stringy is your thing, use bigger chunks of mozzarella as opposed to shredded.
Also, I just checked the private label/supermarket brand of mozzarella that I have in the fridge
30 g serving
5 g fat
1 g carb
7 g protein
The package I have happens to be part skim. I believe whole milk mozz clocks in at 7 g fat.