In all technicality, Descartes did extend his proof beyond I think, therefore I am. He started by assuming nothing, and that was merely the first step. In fact, "I think therefore I am" was stated by a lot of people long before Descartes, including Plato, or maybe it was Aristotle I forget. Augustine by any means. Anyway, if you accept Descartes' logic, then he did eventually come to the conclusion that everything exists.
Again, I don't see any particular usefulness to this definition of existence, but I certainly don't see how or why under that definition, thinking is sufficient to guarantee "existence". If we are asserting that it is possible for the world not to exist, why is it somehow possible that thinking must mean we exist. Thinking doesn't even guarantee free will. Why thinking? Why not Blufferbidging? I blufferbidge, therefore I am. If you do not blufferbidge, very likely you are not. Further, how can anything "exist" in a world that does not exist. If we accept this "I think, therefore I am" model (why should we?) then our next logical assertion should be that for us to exist, the world also needs to exist. For us to exist, we must be biological organisms, and if we are biological organisms then it follows that there must be other biological organisms like us. If we are the only thing that exists, then it follows to reason that the world must exist, or we are God. If we assert that the world must exist, then the objects we perceive within that world should be assumed to also exist - it makes absolutely no sense to put the onus on proving that we exist - the onus is on someone to prove we don't. It's like in a trial. Most people are not murderers. The accused should not have to prove that he is not a murderer, the accuser should have to prove he is.
And I don't think The Matrix is a very good example, because (A) it's not very believable and (B) the people living in The Matrix are living real lives ~ they're not fake lives - they are not fake people, and the matrix is not fake - the matrix is real, even if it is constructed, and the "real world" of the matrix is no more or less real than the matrix itself. The underlying assumption implied in saying the matrix is not real is that unless something is biological and locatable within space-time, it is not real. But I see no reason why this should be the case. If you really wanted to make the argument, you could say that the entire universe is itself a program created by God. But this definition of existence to which we are trying to measure ourselves is useless because it can never be satisfied. Nothing, ever, can possibly be said to exist under this definition, not self or world because you can always draw it back another layer.