The Sublime Text API doesn't have much support for working with projects, so this request might be pretty difficult to implement... I'll look into it but it might be more work then it's worth.
This is already possible if you add the following to your Settings file (Preferences > Settings - User):
"selector": "text.html, source.css"
(The '<' part is from your default settings so you don't want to override that.) Essentially, ST2's autocomplete will activate when you type letters. This setting gives exceptions, other characters that also trigger the autocompletion popup. As for why ../images/backg[CTRL+SPACE] yields no suggestions, this is because the plugin checks if the previous character is a /. If the autocomplete is already up though, this is not a problem.
Hope that helps!