Using Structures instead of Singles
I'm doing this at the moment. It's not a perfect solution, but nor is it terrible. In a lot of ways, I'd quite like Craft to just burn Singles and Channels and use nothing but Structures. You can do everything with Structures that you can with the other two, and you get better flexibility and much nicer groupings in the CMS that make more sense to clients (the one confusing bit of Craft clients seem to have is why editing pages is spread out over three areas they don't see any real logic behind). But, that's not how things are. Anyway...

If you're getting "entry does not exist" then you've got something broken in your routing. Every page inside a Structure is an Entry, and if you've got URLs resolving to entries in your Structure then you've got access to the entry variable in your template.

Check your Structure and see what template it's actually loading. The only thing I can think of at the moment with the little detail you've given is that you've got templates in a path that match the Structure, and instead of routing theough the database it's routing theough the files.

i.e., if you've got templates/my-structure/my-entry.twig and also an Entry with a URI of /my-structure/my-entry then you might be skipping the database because your filesystem is matching first.