You’ll find a lot of Italian chefs used julienned mozzarella/fior di latte, or batons, rather than big circular slices. There’s no firm rule that I’m aware of!
That said, trying to grate fresh mozzarella, instead of low moisture mozzarella (like is generally used on non-Italian style pizzas), would likely leave you with a big wet smeary mess.
I make a dozen large pizzas a week (I do a weekly pizza night for our small cottage bakery) and I cut my fior di latte into strips, then let them dry out a bit, covered on a wire rack in the fridge overnight before using.
Pre-cooked chickpeas are going to be too soft for falafel unless you add a lot of binders; otherwise they’ll fall apart in the fryer. Soaked (but raw) chickpeas are the usual method.
I have had great luck with this recipe: https://www.themediterraneandish.com/how-to-make-falafel/#tasty-recipes-10436-jump-target