Does cheese keep well if it is frozen?!

Hard cooking cheeses like cheddar, swiss, monterey jack, parmesan, etc. freeze well and can be used straight from the freezer. I ususally grate them and store in vacuum bags. Creamy cheeses do not freeze well at all.

I always keep the majority of my cheese block in the fridge, but I will grate a lot and keep in a zip-lock bag in the freezer.
Ours comes shrunk wrapped and has an 'easy open' end. Peel away a little so you can slice some off, then wrap the end around again. It will slow down the development of mould. If you do find a bit of growth still slice and trim the edges. If you open it up and cut away the mould you will find you are forever whittling it away every time you open it.

I have found that when I tried freezing block cheese it tended to crumble after thawing as I tried to slice my way through that part.

The composition of the texture will change, but it freezes good enough not to bother the taste buds.
Yes, shredding is an excellent idea, since when you freeze whole and try to slice the thawed piece of cheese, it will no doubt crumble (hence the texture change).
I have a Vac Seal Bag thing and most of what's in my freezer are in those sealed pouches. For me, it has been one of the best things to keep things 'fresher' in the freezer.
If you use Tupperware, the extra air in the container will add to a freezer burn situation. When freezing cheese or anything, really, it's best to extract most of the air from the vessible which the product will be stored in to reduce freezer burn.

Cheese keeps well enough on its own... cheeses with wax coating last even longer.

It really depends on what kind of cheese you're talking about. Plastic cheese and normal generic cheese don't have a very long shelf life.