1) The soap would refer to a specific instance of the soap being discussed, whereas without the article it would refer to the soap brand in general. The same goes for 3.
2) To refer to Coca-Cola drinks in general, indeed you would not use any article.
4) Using "the" would again refer to a specific instance of a specific product in the Jar brand, but "a" would refer generally to any product in the Jar brand.
For the last examples, the is used for the Bush administration because a specific thing is being referred to – there was only one (or two depending on how you look at it). "Bush administration" is not a class of objects (thank God). "New York City" as a whole is a proper noun, and so there is no article ("City", whilst an optional part of the name, should still be capitalised). Your examples with the book are correct, but the former is by far the more natural. I hope I have helped you.
[Not a teacher]
Interested in Language