[Serious] Could an omnipotent and omniscient god prove that he was God?

[Objectivist]

I disagree with everything that you said but I'm going to confine my response to this one point, since it answers all the rest. 

It is axiomatic.  It's the formal recognition in the form of a philosophic principle that things exist, that there is a reality.  You ask what categories are included in the concept 'existence'.  All of them.  Everything that exists is included in the concept 'existence'.  Whether you find this vague or useless is irrellevant.  It identifies a fact of reality and that's what concepts and principles do.  Do you know what a principle is?  A principle is a conceptual identification of a general truth that other truths rest on.  Knowlege is hierarchical.  The axiom of existence occupies a very special place in the hierarchy of knowledge. The axiom of existence is the widest of all possible truths which all other truths rest on.  In the act of recognizing this most fundamental fact, we grasp two other facts, if only implicitly; that something exists that we are aware of and that we exist possessing consciousness.

  The very first question to be resolved in all of knowledge arises from these two recognitions.  What is the relationship between a conscious subject and its objects, an object being anything we are aware of or consider? That question has to be answered before you can go on to learn anything else about existence.  All knowledge is a mental grasp of an object by some subject, therefore the orientation of the subject-object relationship is a general truth that all other truths rest on.  That orientation is directly observable.  The objects of consciousness have metaphysical primacy over the subject of consciousness.  That means that things are what they are and do what they do independently of consciousness.  Anyone can test this at any time and at any place.  Pick any object in the range of your senses and think about it being different.  Think about it rising in the air and twirling around.  Does it obey or does it remain what and how it is?

These are those truths that we can be certain of because without them no knowledge is possible.  As soon as anyone says "it is" they are implicit.  It (existence) is (exists) and since this is a statement of knowledge (consciousness), the subject-object relationship is also implicit.  These truths are necessary and they are inescapable.  These truths are the defeater of the notion of the Christian God.

As the base of all knowledge anything that contradicts one of these principles can not be true.  They are the standard by which all truth is judged.  The notion of the Christian God violates all of them.  Therefore, I can know for certain that the claim that the Christian God exists  can not be true, and is self contradictory.   

Existence exists and consciousness exists.  Consciousness is the faculty that perceives and identifies what exists but does not create or alter what exists.   If you reject these truths then you have to reject all statements of knowledge but even this would assume the very thing rejected because if existence doesn't exist then there's nothing to reject and no consciousness to reject it.

I'm still having a hard time understanding how "existence exists" explains why/how particular things exist. Yes, existence is real, but so what? How is this more explanatory than saying God exists (for example)?

For example, how do you get from "existence exists" to "this local universe exists". How does a principle do that exactly?

It doesn't.  That's not it's job.  It's job as I said is to identify a general truth on which other truths rest.  How you get to further knowledge is by the same process of looking at reality and identifying what you perceive.  We don't just recognize that existence exists and deduce everything else from that recognition.  That's not the way knowledge works.  We learn almost everything inductively.  Only after we've done induction can we apply that inductive knowledge to particulars.  We can't say existence exists and then jump straight to why/ how does the local universe exist.  What is a universe, what does local mean?  What is causation?  Before we get to those concepts we've already made a long chain of discoveries that lead up to those questions and when we get to them we answer them, if they are not improper questions, by looking to reality.  We don't start with nothing and then seek a reason for why existence exists.  You have to start with existence and then see what else you can learn about it.  The question of what came before everything and what caused it is an improper question because it makes use of stolen concepts.  To steal concepts is to make use of them while ignoring their roots including the recognition that existence exists.