Since the World Wide Web is a proper noun, we use initial upper-case letters, as we would with your surname, for example.
As for writing ‘Web site’ as one word, it is true that this is seen a great deal, but then, so is the spelling ‘recieve’ which is just plain wrong!
I disagree. Languages change over time, and I think 'website' has now become the accepted spelling. A Google search for 'website' returns almost 3 billion results, including the official website of the British Monarchy. If 'website' is good enough for Her Majesty, it's good enough for me :-)
What do you think?