How to set a static background at wordpress theme?

If you set your theme to the twentyten template, go to appearance > Background.
Upload an image, you can then have it centered and fixed.

Once you’ve done that, you’ll see that the image is like the one in the example. Just look at the css and you can use something similar in whatever theme you want.

You can also use a plugin like this one – http://wordpress.org/extend/plugins/wp-supersized/
to get the image to be the whole size of the browser.

As for the snow, not sure, but if you look at the example site, just look at the source code. It’s all in there.