Man is a metaphysical animal. We philosophers ought to encourage this tendency in our fellow mortals. This morning's mail brings me a long disquisition by a Spanish lawyer (abogado), Daniel Vincente Carillo, entitled "A New Argument on the Existence of God." It consists of numerous definitions, axioms, and theorems. I don't have time to comment on the whole thing, which can be found here, but I will remark critically, and I hope helpfully, on his modal axioms.
You are of course right that nothing is both contingent and impossible; but this does not show that Axiom 5 is incorrect. "What is contingent is possible" is true; and from this "What is contingent is either possible or impossible" follows. If, as I gather from your account, Carillo denies that the class of impossible contingent things is empty, he is mistaken, but his axiom isn't. One can object to it that it isn't an axiom, as its truth depends on the truth of "What is contingent is possible." This is probably too trivial a point to have written to you about, but I pass it along anyway.