Why do Athiests require 'proof' that God exists?

[Ryft]

It seems to me that whether or not God exists a person's understanding of rationality ultimately comes from [the] universe.

First, perhaps it "seems" that way to you because, as you admitted, this is an area that you have not properly studied. Second, your statement follows only if granted the assumptions with which you approach the question; but they most certainly are not granted—nor should they be—since those assumptions regard the very question itself. (It is illegitimate to simply assume the very thing to be proved.)

The blue ball cannot be the red ball.

But your observations do not support that conclusion, Tegh. What they support is that the blue ball is not the red ball, a descriptive fact of ‘a posteriori’ knowledge. And your further observations from trying to "mash them together" as hard as you can confirms that "they are still separate entities," that "the red ball is still itself and the blue ball is still itself" (emphasis mine), which again are descriptive facts (is and is not) of ‘a posteriori’ knowledge. But to say that the blue ball cannot be the red ball is an altogether different thing, a normative fact (can and cannot) of ‘a priori’ knowledge. You do not succeed in bridging the deductive gap between these two distinct categories (descriptive and normative) by conflating them, for that is a basic logical error.

Perhaps this is ultimately where the law of non-contradiction comes from.

No, and for the above reason. The law of non-contradiction is normative, not descriptive.

It's inconceivable for the red ball to be the blue ball because I have no experience of this being possible.

First, one's inability to conceive of X has no bearing on the truth of X; that is an argument from incredulity, which is a logical fallacy because it ignores and fails to eliminate the possibility that something can be both inconceivable and nevertheless true. Second, the law of non-contradiction is normative, objective, absolute, and necessary truth, by which I mean that it is not descriptive, subjective, relative, and contingent (such as what your mind is able to conceive). It is not as if the universe was existing and behaving contrary to itself for several billion years until philosophers finally arrived on the scene to put a stop to it with the law of non-contradiction. Reality itself is ‘non contra se’ quite apart from our existence.

Also, I don't see how positing the existence of God helps in this matter.

I already answered that.

If all our knowledge comes from experiences and memory of the space-time universe ...

It does not. We also have knowledge apart from experience (‘a priori’).

... and God is [transcendent and] eternal, perhaps he could have created a universe made up of something other than space-time, and we would then have completely different "logical" laws.

That is a contradiction. If God eternally decreed to do X, then it contradicts the eternal nature of God to say he could have decreed to do ~X; in other words, to say that God decreed to do X but could have decreed to do ~X is to imply temporality (mutable), not eternality (immutable).

(P.S. We are at risk of hijacking this thread, which is supposed to be about why atheists require proof that God exists. If you want to pursue this further, then perhaps you might consider starting a new thread.)