Ryan, doctype is supposed to instruct the browser which HTML standard to use. HTML5 doctype says — do whatever you want.

Re: header background. I agree that inline CSS is not the best solution, but it certainly the easiest of the semantic ones. WordPress doesn’t have a built-in mechanism for generating and caching dynamic stylesheets which would be ideal for this kind of a theme feature.

Re: widgets & lists. There is a better solution and it doesn’t include unordered lists. Just choose anything else and it’ll be better than lists ;)