Isn't that precisely what it means? You are aware of something, thus you can take it and its aspects into consideration in your calculations. You have a representation of yourself - same thing. Of course your representation may not be completely accurate, but what perception would be.
I don't think so. The concept of self is distinct from an image (or other representation). If I run pylint against the pylint source code, it has a representation of itself. It's not aware that those lines of code are in any way special to it. I don't think it mystically becomes self-aware because of that situation.
In your lingo, something can know what it looks like without taking its aspects into consideration.
I think the inverse is true, though: something that cannot perceive a representation of itself cannot be self-aware.
Maybe or maybe not. This is the nub of the debate. I would argue that answering "yes" to this is a partial endorsement of pan-psychism. If the ability to experience qualia is property of certain algorithms or types of information flow then it's a fundemental property of the universe.
Thought experiments about p-zombies and mind simulation are an interesting litmus test to separate different points of view on this.
Having a representation of itself doesn't mean it has an awareness of itself.