Well, I cant prove to you that I can think, but the very fact that im considering this argument shows to myself that I can think. In order to think it is logical to assume that I must exist (in some format).
=> "I think therefore I am"
Your second point is highly dependant on the definition of "exist". My definition of somthing that exists is "an object (no specified format) that has the potential to interact with me". Under this definition, if an object does not exist, it will not affect me and whether or not it can think becomes a moot point.
EDIT: On second thoughts, by thinking, I am interacting with myself, therefore under this definition I exist.
*google's solipsist*