Yes it breaths, it's cotton.
As far as waterproof goes, it's not really meant for use above freezing thus any precipitation is snow. I've had snow build up on it but not melt as I brush it off. It isn't waterproof, another reason why it's used in layers, primarily wool. Wool retains it's heat retaining properties when wet, better than any synthetic ever devised. It also doesn't burn like synthetics so camp fires don't leave pin holes in it.
An Anorak is not meant to be used on it's own. Staying warm in the winter while spending large amounts of time outside is only possible when you use a system of layers and materials. The common misconception with people who've not done a lot of outdoors activity in extreme cold, they think thick, big, heavy coats and parkas. That's ok for sedentary ice fishing or something but not for activity. They are bulky, cumbersome and over heating if you are doing any work at all. If you do commence heavy activity in these sort of garments you will sweat and when you sweat you're really getting into dicey territory for hypothermia. A system is required for comfort in extreme cold, you won't find it in singular garments.
The phrase cotton kills is so misused it's not funny. Cotton will kill you as a base layer as it does not move moisture away and creates a cold sink on the skin bleeding you of heat. As an outer shell it works because it lets heat out. Anoraks like that are to be worn over layers, it isn't an end all outer layer.
A cotton anorak will load up with moisture absorbed from the heat your body is producing, pushing all that moisture out and through your inner fleece or wool layers. One of the things I do is squeeze my arms into my chest when I start to get too hot, this pushes that internal heat out of the garment. I know it sounds crazy to say you need to get stay cool to stay warm but the concept is critical to understand. On high humidity days the anorak will actually get wet, even when well below freezing, and the fabric will itself freeze. That's absolutely no problem since its on the very outside of your system. It stays pliable because of its fabric properties and your body heat. Even when frozen the moisture will sublimate. Around the fire it dries as you wear it, and you will see the steam come off it. Staying warm isn't so much about keeping all of your body heat, as it is keeping the right amount of body heat. Too much and you sweat causing moisture build up, which then steals your body heat faster than you can generate it. Only a system of clothing can help regulate that heat exchange.
For anyone considering an Anorak or just looking for more knowledge about winter trekking and activities
www.wintertreakking.com is one heck of a great resource.
For more detail on why an Anorak and layers are superior
here is a great break down of the benefits and proper use.