Um, are you sure you're not a religious troll trying to discredit atheism?
Everything here is so bad it's embarrassing. If this is an honest attempt...I'm sorry, but please do something to sharpen your thinking skills before trying to make contributions that will actually enlighten people? Philosophy classes are pretty good for this, generally.
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Absence of evidence can be evidence of absence; in a Bayesian framework you can even condition your other hypotheses on the absence of evidence and among other things you'll see how much your predicted probability of actual absence will go up.
However, you can't just leap from that to stating a proof. It's all inherently probabilistic.
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It's also true that you can prove an absence of things via logic (including modus tollens). However, your premises need to be true in order for the argument to be sound.
But you don't justify your premises, so you have something that is valid logic but not a sound proof.
And the problem: one of your premises is "there is no such evidence". To justify this it seems as though you need to...prove a negative.
Oops.
Not too bad, but really not convincing, and really not good to be unclear about whether you have supplied a sound proof or just a valid argument.
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Your mathematical proofs fail badly for multiple reasons, but mostly because of lack of justification of premises.
For instance in your Algorithm Box proof, you provide not even a hint of why God's algorithm box needs to be using a "ω-consistent recursive axiomatic system". Even we don't need to do that. How do you know God doesn't just have access to the equivalent of a lookup table, for instance? (And "all knowing" from our perspective might be different from "all knowing" from God's.)
The proof of non-existence via non-omnipotence is even worse. You make oodles of unsupported assumptions like power is quantifiable, "omnipotent" is equal to not just infinite but "the largest infinity" (reasoning: "of course") and then...you state by fiat that no such thing exists. You don't actually say how that was proved, but if it's in ZFC...well...how do we know we're living in ZFC? If so, wouldn't like the entire von Neumann universe do the trick?
Anyway. Not a proof.
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You can prove a negative. It's just hard.
There are two methods.
Method one: exhaustive search. You look everywhere that X can possibly be, and there is no X. For instance, I can prove that there are no letters 'm' in the following quoted phrase: "This is a quoted phrase." You examine every letter. No m. Okay, cool.
Usually it's impossible to search exhaustively, so usually this doesn't work.
Method two: logical proof. You can prove using any of the usual methods--by construction, by contradiction, whatever.
For example, here's a proof that there are no integer a and b such that a/b = sqrt(2).
Assume there are, and reduce a and b so they have no common factors. (I won't rename them--just assume they're reduced.)
a = b*sqrt(2) so a^2 = b^2 * 2.
So a^2 is divisible by 2, because it obviously has a factor of 2 that b^2 doesn't. But that means a must be divisible by 2, so a = 2*k for some k.
a^2 = (2*k)^2 = 4*k^2 = b^2 * 2. Cancel the two
2*k^2 = b^2.
So b^2 is divisible by 2. So b is divisible by 2.
Uh-oh. a and b are both divisible by 2...contradicting our requirement that they were reduced until they had no common factors. Therefore, there can exist no integers a and b such that a/b = sqrt(2).
Okay, very good, we have a proof of the absence of something. But it was a gosh-darned actual proof, with axioms providing our true premises.
For claims about real-world existence, it's incredibly difficult to show that your premises are true, if you manage to construct an argument that would prove something doesn't exist. (Usually the best you can do is prove that given a certain model, something can't exist, but you don't know whether the model perfectly reflects reality.)
So it's technically possible to prove a negative, but practically the demands are super-super high.
You have fallen immensely far short of doing that here, so my guess would be you probably haven't succeeded anywhere you've tried for an interesting result (like "God doesn't exist"). You might have proofs of some weird things like "God, meaning this conception I have of God that doesn't necessarily match anyone else's, doesn't exist".
Anyway, you have a few clever ideas, but you need way way WAY more rigor...unless this really is just a type of trolling. (If Christians hired Russian information ops people to help promote their beliefs, the Russians would totally do stuff like this.)