Your “translations” are incorrect. The logical formula A → B means, “If A is true, then B is true”, or “A implies B”, or “B follows from A”. Substituting G for A, and G for B, we have G → G. That means effectively that if God is to exist in actuality (G, which is to be proved), then God has to exist necessarily (G, which is how God is defined.) The logical formula (A → B) means, “It must be the case that… etc.” (See the description for “A → B”.) And indeed, it makes sense that it must be the case () that if (->) God does exist in actuality (G), He has to be what we defined Him as: (G). This is the least controversial premise and is in fact self-evident. To take a more mundane example, say that we defined Bob as a man who wears a red shirt. Since we cannot prove things simply by defining them, it stands to reason that we must establish that the man we have defined does indeed wear a red shirt. Otherwise, it could be the case that Bob wears a blue shirt, and we have redefined “red”. Therefore, a reasonable premise would be: “It must be the case that if the Bob we have defined does wear a red shirt in actuality, then he is the Bob we have defined who wears a red shirt.” Number 7 is a direct inference from number 2, and as an inference whose modality is following a valid rule of logic (modus tollens), the inference cannot be anything but valid. We don’t have the luxury of saying that a rule may be applied when it suits us, but not when it doesn’t.
The only way to modify this particular proof, to say that God does not exist, is as I explained above. You must reject the first premise, ~~G (it is possible that God exists) and replace it with the premise, ~G (it is possible that God does not exist). You may then follow the identical tableau, and your conclusion will be ~G (God does not exist). But — and speaking of untenable paradoxes — if you substitute the definition of God for the term “God” in the premises, you can readily see that one makes sense and the other does not:
(A) ~~G = It is possible that the being that exists in all possible worlds exists.
(B) ~G = It is possible that the being that exists in all possible worlds does not exist.