Wow, I keep finding these threads that I should have found right away. I started making cheese almost 2 years ago. And beer only in january.
For a softer mozzarella I found that it is really hard to be done with the microwave and knead/pull method. The best way is to strain your curd. Heat your whey to about 180. Press the curd into a small ball and dunk into the hot whey. Less whey will be lost. After 30 sec to 1 min strain out kneed and dunk again. It usually takes a few dunkings until it gets glossy. Then pinch off what size ball you want and dunk in ice water. Afterwards you can also salt the whey and use that to store your cheese in. The softest mozzarella I ever got I added 1 cup of 1/2 & 1/2 or heavy cream to whole milk and took care not to kneed too much, then stored in olive oil. Ended up too soft, that I was gaging on it!
I have only done a couple of batches, but I found that over-kneeding really made the first batch tough and chewy. The second time I microwaved and heated more carefully and it turned out better. I also under salted the first batch, and the second was very tasty. I think having more fat content will make it softer as well.