Depends on the cheese, and on the usage, really. My mom works at a cheese factory, so I regularly get more cheese than I know what to do with, so most of it ends up in the freezer. Non-aged cheese (not familiar with the English terminology, sorry) is absolutely no problem to freeze in my experience. Aged cheese, at least the ones I eat, crumbles completely after defrosting, absolutely impossible to slice - but I only use aged cheese for pizza and other stuff that goes into the oven, so the fact that it falls apart just by touching it is more of a feature for me - no need to grate it!

It depends, I buy a large block of cheddar, cut off a weeks worth and freeze the rest. The frozen cheese of course does not retain the texture but is fine for meals calling for melted cheese. Cheddar usually comes in quantities that would mold if kept in a fridge until consumed for me, so it’s either waste excess by letting it mold or having meals with melted cheese for a few weeks.