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

[GrandizerII]

As I recall, “Existence exists,” was considered axiomatic by Rand but I find vague and useless as premise. What exactly categories of being does Rand include in existence? Physical bodies? Sets and numbers? And Rand dashes off to fight communism before dealing with the complex questions such as if there is a distinction between Being-As-Such and Being-In-Itself, etc. Does this particular physical reality exhaust the fullness of what is, as just an incurious brute fact, or is it a manifestation of one possibility out all possible worlds within a larger reality that includes necessary and true transcendent principles and powers across all possible worlds. My vote is for the larger reality. YMMV.

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?