The index.php file won't show anything in "design view" because it doesn't contain any design. If you look at the source view you'll see that there's nothing there that can be shown in design view.
When you are modiying a theme you need to look at the other files, like header.php and footer.php to be able to see what they will look like. One BIG word of warning with these files though... They are only fragments, not full pages, so they will normally display incorrectly no matter what you do. Some may look OK, but Dreamwaver is not known for it's intelligence when it comes to things like this. Post people use the source view because it's the best way to modify things and keep everything the way that it should be. The design view in Dreamweaver can tend to add in a few extra tags every now and then to "fix" what's not actually broken.
As another semi-related issue... You are doing your modifications in a child theme so that they don't get over-written by any future updaes, aren't you? :)