i'll just address your point regarding "I think therefore I am". Descartes has started from the point of radical doubt, meaning that he is positing the possibility that everything he thinks is reality really isn't and is fed to his senses by the evil demon. Therefore, he can't claim the existence of a table by saying "Look, I see it right there" because what if the table is like a hallucination - it only seems to be real? (The Matrix really helps people understand this part)
At the end of the day, though, he has to admit that at least he himself exists. If he denied that, it would be implied in the very fact that he is denying something. That is, "I do not exist" would be a thought that is occurring to himself... he has no logical way to imagine that he himself does not exist, because by the act of imagining he is acting and thus existing.