Everyone seems to have a different notion of what the idea "God" is, so proving or disproving that idea seems like a fruitless exercise.
Quite true. In fact the Madhyamikas would say that any assertion about something conceived as 'the absolute', whether positive (exists) or negative (doesn't exist) would be meaningless. Accordingly Buddhists generally refrain from speculating about questions of this type. That is why their focus is on meditation and the actual practice of the path. Such debates can and do result in interminable arguments which can never be resolved.
So that is kind of a "Buddhist disclaimer". However there is another answer to Dawkins, from more of a Christian viewpoint. This is that 'God' is not 'something that exists'. 'God' is not some kind of super-being or celestial deity, or, in fact, any kind of existing thing whatever. Of course the human imagination will want to imagine some being that exists, so we came up with:
which I am sure is the kind of being that Prof Dawkins has in mind whenever he talks about 'God'. (Incidentally, it is not for nothing that the Muslims won't allow any images of deity whatever, but I digress).
However, I think the authentic conception or understanding of 'God' is mystical and therefore, not able to be represented in any kind of image whatever. In my view, because Christianity was born out of an amalgamation of a number of different ideas and traditions (Biblical, Greek, Gnostic, and even Pagan), there always was, and remains, a great deal of unclarity about these ideas. But a quote from a text about the theologian Paul Tillich is much nearer to my understanding of the meaning of the word, if it is to mean anything whatever:
"Religion is direction or movement toward the ultimate or the unconditional. And God rightly defined might be called the Unconditional. God, in the true sense, is indefinable. Since the Unconditional precedes our minds and precedes all created things, God cannot be confined by the mind or by words. Tillich sees God as Being-Itself, or the "Ground of all Being." For this reason there cannot be "a" God. There cannot even be a "highest God," for even that concept is limiting. We cannot make an object out of God. And the moment we say he is the highest God or anything else, we have made him an object. Thus, beyond the God of the Christian or the God of the Jews, there is the "God beyond God." This God cannot be said to exist or not to exist in the sense that we "exist". Either statement is limiting. We cannot make a thing out of God, no matter how holy this thing may be, because there still remains something behind the holy thing which is its ground or basis, the "ground of being."
So in this understanding, "God" is not "a being" but "being itself", the source of existence, rather than something that exists. And Richard Dawkins shows not the slightest inkling of that way of thinking. That is even obvious to secular critics of Dawkins, such as Terry Eagleton, who says in his scathing and hilarious review of The God Delusion
Dawkins speaks scoffingly of a personal God, as though it were entirely obvious exactly what this might mean. He seems to imagine God, if not exactly with a white beard, then at least as some kind of chap, however supersized. He asks how this chap can speak to billions of people simultaneously, which is rather like wondering why, if Tony Blair is an octopus, he has only two arms. For Judeo-Christianity, God is not a person in the sense that Al Gore arguably is. Nor is he a principle, an entity, or ‘existent’: in one sense of that word it would be perfectly coherent for religious types to claim that God does not in fact exist. He is, rather, the condition of possibility of any entity whatsoever, including ourselves. He is the answer to why there is something rather than nothing. God and the universe do not add up to two, any more than my envy and my left foot constitute a pair of objects.
If you're interested in a really informed analysis of the controversy, from a generally spiritual-but-not-too-religious perspective, from someone who really does understand the subtleties, have a look at Karen Armstrong's The Case for God
Anyway, this is a Buddhist forum, and, fortunately for Buddhism, it doesn't really have a dog in the fight.