Eiuol, do you mean claim of the existence of God is arbitrary or that coming up with the idea of such a thing is arbitrary or both? I think you mean the first. Although, I'm pretty sure that a Scholastic philosopher, say, who thinks they have a good proof for the existence of such a being and its principal characteristics like and unlike ours doesn't think anything at all is arbitrary about it. My first philosophy professor, as you know, was a Thomist. He began his proof for the existence of (and some character of) God by holding up his hand and having us observe him crooking his little finger. He would then trace back causes from that alteration, invoking various philosophical concepts and propositions along the way, and though I did not know of any an the class who were persuaded by the elaborate argument, I doubt the professor could see anything wrong at any step of the argument, and surely not anything arbitrary.
Eiuol, we can go further. ". . . idea of God is that God is a somewhat human-like being, a personage." Rather than this God making humans in his image, as the Genesis account had it, Rand had it (in AS) that humans made up God in their image. I think she is right. Believers refer to it as a him, indeed as a father or a son. For a great many, it is a constant human-like companion, just like say an absent loved one present in all one's moments of consciousness. Popular songs say "God is watching us"* as a person-like intelligence and "he walks with me and he talks with me and tells me I am his own."* Much human.
We do not capitalize God out of reverence or out of respect for someone else's reverence. We capitalize it in general secular culture as a proper noun, as David has mentioned. From the Chicago Manuel of Style:
"Like all proper nouns the names of the one supreme God (as Allah, El, God, Jehovah, Yahweh) as well as the names of other deities (Astarte, Dagon, Diana, Pan, Shiva) are capitalized.
"The one God. Other references to deity as the one supreme God, including references to the persons of the Christian Trinity, are capitalized. . . ."