Without seeing your HTML and CSS, it's impossible to say why it's not happening. It's almost certainly due to an error somewhere in your code.
I have written a much more detailed tutorial series based on the same design. It goes into exhaustive (some might say exhausting) detail with step-by-step instructions on how to build the site. You might find it helpful in understanding how Dreamweaver CC works with HTML and CSS. It's a six-part series. Part 1 is here: http://www.adobe.com/devnet/dreamweaver/articles/first_website_pt1.edu.html.
The CSS code I'm using is in your main.css file which was included with the tutorial files. I did not do anything with the links, other than the steps you mentioned. When I got to the zeroing out section, that's where the problem occured.
Actually, it appears that the Padding control is not working properly. Whenever I restart Dreamweaver CC and go back to main.css and select #mainnav ul, or do something else and then reselect those last mentioned items, the Padding box has only a 0 on top and the all the others are grayed out. This occurs even though I have clicked on the chain links to connect them and clicked on the top 0 to select it and then clicked away before and then all of the Padding box sides have zeros.
So, Dreamweaver is not keeping the setting, which I had just set to all zeros on each side earlier but only on the top side.
Because of this problem, I cannot go any further with this tutorial! I just tried to look at the Style the navigation bar session, again and I cannot view it, probably becasue of a downloading issue, which may be caused by an overloaded server.
If Dreamweaver is not keeping the setting, it means that you haven't saved the style sheet (File > Save All Related Files).
Looking at your screenshot, the link in the centre of the padding tool is in the default state. This is what it looks like before you start:
When you click the chain icon, the links join together like this:
When they're joined together, you can set all sides to the same value by changing just one.
If the link in the chain icon is broken, each side must be set separately.
Yes, I am familier with this. When I went to File-Save all Related FIles, it was grayed out as if it was already done. If I make the selection so all sides of the padding box are zeroed out, with the chain links connected, and save the file using Save all Related files, and then reopen Dreamweaver and go back to where I was, the padding box only has a zero on the top side as shown in my earlier screenshot.