Honestly, I don't think you are listening to us. Let me try to distill this to a simpler set of statements.
Any link that is root relative will NOT preview properly in the browser UNLESS you have configured your box with virtual hosts.
All links that are root relative will display properly when the files are uploaded to the server (assuming that the path in the link is correct).
Your problems are not related to browsers or browser versions.
Think about it. When DW hands the file to be previewed to the browser, it has no way to tell the browser where the root folder of the local site is. So when the browser sees the leading "/" it begins looking for the linked file at the root of the hard drive, not the root of the site. This preview will fail...