You're wrong. You're arguing from surface similarity rather than detailed internal workings, which is a big no-no in maths. I could have spent about five paragraphs breaking down your argument point by point, but you've made my life easy with that final line:

therefore it is able to evaluate everything needed for Godel's proof, therefore it must have not only one but an infinite number of cases for which it hangs.

This is not the case, there is a simple counter-example. Unary code, in which 0 is 0, 1 is 10, 2 is 110, 3 is 1110, 15 is 1111111111111110, hangs on the string 111111.... and no other. The fact your argument led to this conclusion is a demonstration of how completely wrong it is.

You are right, in that those statements do satisfy the conclusion, and I don't see why you're being downvoted. The difference between them and the original, self referential, statement is generality. For example:

You: I know that ZF set theory is incomplete because the axiom of choice cannot be proven within it.

Some other guy: Okay then, how about we add the axiom of choice as another axiom, maybe this new system will be complete?

You: Nope, it still can't prove or disprove the continuum hypothesis.

SUG: So I'll add that in as another axiom, maybe that will finally patch it?

You: Nope, still doesn't work because ...

SUG: But if I add that as an axiom as well...

Hopefully you can see why this might keep going for quite a while, and given that its quite difficult to prove a statement is undecidable you will run out eventually. There's nothing you can do to convince him those undecidable statements are symptoms of a general problem rather than just one-time flaws.

Compare to this:

Godel: ZF set theory is not complete, because I have this self-referential construction of an unprovable statement.

SUG: But what if I add your statement as another axiom?

Godel: Then my proof still applies to this new system you created, and I can construct another, similar statement.

SUG: But what if I...

Godel: There's no point in you trying to continue this, whatever system you create my theorem will always apply.

SUG: Drat! Foiled again!

Those are statements that fall under what Godel proved, that is they are statements that are unprovable in ZF. So even though his statement doesn't include self-reference it can still fall under Godel's proof if his decoder is strong enough to determine what is an integer and what is not an integer. Self-referencing has nothing to do with it at all.

The existence of specific undecidable statements in ZF or ZF - AC is a different sort of result than what Godel showed. That for example the continuum hypothesis is undecidable in ZFC is interesting because the continuum hypothesis is interesting. However, Godel's theorems show that any consistent, axiomatizable systemn with all valid proofs recursively enumerable, that is strong enough to model a large chunk of N, must be incomplete. Exhibiting results like Cohen's results about choice and the continuum hypothesis don't give you the full result, they just show specific things about the system ZF. Indeed, if one didn't know Godel's theorems, and just knew about Cohen's forcing results, you might be tempted to just throw in AC and GH as additional axioms if you didn't have them already and then wonder if that system is complete.

It is very funny to me that my most downvoted comment isn't about religion but is about Godel's proof and no one gave a decent refutation of what I said.

People here have a very complicated set of attitudes. Even as many don't want to bother talking about irrational aspects of religion, or engaging in religious individuals who are convinced that their religious view is somehow different from all the other religious views, people who are religious are that way due to a variety of very strong cognitive biases. So there's some sympathy there. Getting math wrong and then insisting one is correct is just simple arrogance or at least, only Dunning-Kruger. There's much less sympathy there. And yes, people have tried to explain why you were wrong albeit fairly succinctly.