Trevor, Dave brings up some good points. When I talk about Walmart cheese not being problematic, it's within the context of supermarket cheeses. When you get into commercial cheeses (such as Grande), they melt far better, so that's why your take home skin was so successful.
When you get into supermarket cheese, I don't see the differences being that dramatic, especially since supermarket brick mozz can vary so much seasonally. Sorrento is consistently better than Walmart, so if you can find that, it might be worth a shot, but, honestly, I'm not sure I'd pay double for it. I would probably pay double for Grande (once in a while) because that's a night and day difference, but Sorrento and Walmart are pretty close.
The consistency and quantity of the sauce greatly define how the cheese cooks. A little more water in the sauce, a little less, more sauce overall, less, and you're looking an entirely different pizza. The differences are surprisingly dramatic. It took me years to dial in my sauce consistency and quantity and I still haven't completely nailed it. Sauce contains water and water takes a lot of energy to boil. The more sauce you have, the longer it takes to boil, the less heat makes it to the bottom of the cheese, the less bubbling. The sauce needs to be wet enough to boil and encourage bubbling, but you shouldn't have too much of it.
Everyone, these days, lives pretty close to a Walmart, and, from my experience, I've never seen their mozz go on sale, so, from my perspective, I don't think there's an advantage to freezing it. I would just buy it when you need it and use it within 2 weeks.
The grater in the photo is your typical garden variety box grater. It usually has a fine hard cheese side, a fine soft cheese side, a coarse soft cheese side and a slicing side. You should be using the coarse soft cheese side (as shown). You might find one or two NY pizzerias slicing rather than grating, but it's only a very small number - maybe 1 in 1000.