To approach your larger question, what are the real reasons people believe or disbelieve, I've offered a bulletized list for anyone who's interested in pursuing this question:
For theists, then:
These, and probably more could be added, are reasons for belief and unbelief. Faith and unbelief, in my experience with people, is generally caught and not taught. The well-considered reasons generally follow; there are notable exceptions, I'm sure, but it's not normative for the well-considered reasons to lead. If you like, we can add, delete, unpack, and/or expand these.
To your specific questions, then:
"That said, I am interested to know more about this feeling you speak of, and I'd like to hear you describe it in more detail, if you can. Is it a unique quale, something indescribable through other sensory modalities, or is it an awareness that comes through the usual five senses?"
That this particular portion of my initial post would have garnered the interest it has baffles me, to be honest. I inserted it as almost an afterthought, because I suspect many theists use this awareness as a basis for God's existence. I do not, nor am I the charismatic type Christian who would be prone to such experiences. I suspect my temperament mirrors yours in many respects.
Nevertheless, we imagine ourselves separated by a gulf of experience, so let's press on the best we can. Can I describe this awareness to you in more detail? I doubt it. The closest I might bring you to the experience is your encounter with the sublime or perhaps the numinous, so let's take a quick look at both.
Certainly you've encountered the sublime: a gaze at a sunset, a fascination with the stars, a sense of something greater than yourself. In fact, I believe I recall your exposition of the sublime from an atheist's perspective in one of your essays. I'd not suggest to you that your confrontation with the sublime is equivalent to the awareness I've mentioned. It's not; however, theists tend to meld the two in their minds, so perhaps that experience of the stars at night is as close as I can guide you to my personal experience. I suspect it is.
But, perhaps the numinous, a term coined by Rudolf Otto as far as I know, is more fertile ground. Otto described the sense of contact with a being wholly other as the numinous. While I would not describe God as wholly other — there must be some common frame of reference for contact with God if we were to know him — the conception of a being similar to the attributes customarily ascribed to the Christian God should engender a sensation of the numinous. The feeling produced by the holy God described by Christianity may cause this aspect of Otto's numinous: the mysterium tremendum, an unsettling awareness, one perhaps of fear. Moreover, there's the mysterium fascinas: as the phrase suggests, an awareness of a being so infinitely wonderful that it's irresistible in its allure.
Hopefully, that gives you an inkling of the experience. It's an odd situation. I have no doubt of your honesty when you claim to possess no like experience, yet I'm certain that billions of theists would report similar experiences. They'll know what I'm talking about, but collectively we won't be able to adequately explain it to you.
In that manner, it does resemble a quale, doesn't it? But I hesitate to term it such, for it ushers in a host of philosophic associations that may or may not be helpful, and they may very well prove misleading. I also hesitate to utilize the conceptions of sensory modality and the usual five senses. An historic theological phrase, the sensus divinitatis, is more than likely the best descriptive vehicle, but it carries baggage when used around atheists that I'd rather not unearth, as I've stated previously on DA. What I can say — for myself, that is — is that it appears to be part of an epistemic cognitive function capable of apprehending this awareness.
But, of course, this last statement is contingent upon the de facto consideration of whether God exists. If He does not in reality exist, then your (and mine, actually) likely conclusion that I have a God gene or some other neurological peculiarity, as you put it, seems almost certain. That, or I'm simply deluded. Either way, it would seem that here I stand, I can do no other, unless of course you are successful in convincing me that God does not in fact exist, which may not prevent the awareness, but only provide me a better explanation for the phenomenon. Naturally, another option is that God actually exists, and this awareness somehow is reflective of an actual presence. And, if we care at all to logic, it would appear that there may be other possibilities available to us as well: perhaps God exists and this awareness is in no way related to him. Whatever the case may be, the question is bound inexorably to the de facto question of existence, so while it may be interesting to ponder, it seems to me it has to be tabled until the time that question is actually settled. Until that time, if there is one, the theist and atheist are likely to proceed with their thinking in relation to this question based upon their current beliefs.
So, then to your second concern:
"Why is it the case that justice, consciousness and the like raise the odds in favor of a world-with-God hypothesis over those of a world-without-God hypothesis?"
As you well know, this question, and any subsequent answer by a Christian, will mire us in the invariable discussions endlessly volleyed by Christians and atheists. And it leads the theist inexorably into an axiological argument for God's existence. For example, I'd be interested to know based on your description of the world as you see it:
"It's easy to see how those good things you mention could come about by accident, at least some of the time, in a world with no higher authority; random chance will sometimes turn out in our favor, sometimes not. But I think it's a lot more challenging to explain how evil and injustice could come to be in a world overseen by a deity that does not desire such things."
how you would ever conclude that there is evil and injustice. If these things come about by accident, as you say, why would we consider them good? If they come about by random chance, where's the injustice or the evil? Certainly you don't conclude that there's evil and injustice in the insect world, yet if we're the same product of naturalism that the insect kingdom is, and there's no higher authority overseeing our existence, why would we presume that there's actual injustice or evil simply because we're a more highly evolved lifeform with an emergent consciousness? Did we awaken in this world as Gregor Samsa, as monstrous vermin?
But before we do all that, let me address the greater question of the problem of evil:
"But I think it's a lot more challenging to explain how evil and injustice could come to be in a world overseen by a deity that does not desire such things."
We need to frame this question before delving into it. Many atheists, not to suggest yourself, are unaware that the logical problem of evil is now, I'm pleased to report, widely abandoned. The logical, or deductive form of the problem of evil attempts to demonstrate that the propositions "God exists" and "evil exists" are contradictories. The primary cause of this wholesale withdrawal has been the inability for philosophers to demonstrate that God cannot possess a morally sufficient reason to permit evil. Hence, there exists no persuasive deductive path to demonstrate successfully a contradiction between the existence of God and the existence of evil.
For instance, the highly esteemed atheist philosopher, and former DA poster, I believe, Dr. Michael Martin has stated "Most philosophers now believe that there is good reason why the Deductive Argument from Evil fails: it is logically possible that evil can exist even if God exists if God has good moral reasons for allowing it." Moreover, atheist philosopher William Rowe states "Some philosophers have contended that the existence of evil is logically inconsistent with the existence of the theistic God. No one, I think, has succeeded in establishing such an extravagant claim. Indeed, granted incompatibilism, there is a fairly compelling argument for the view that the existence of evil is logically consistent with the existence of God."
While this is inconclusive in itself with regard to whether the problem of evil is a true defeater for God's existence, I think it is important to note that there's no logical or deductive path between the existence of God and the existence of evil that impedes belief or founds unbelief. Thus, the problem of evil is relegated to inductive or abductive arguments.
In fairness, then, I would expect every atheist to approach the POE with the same level of skepticism they showed with my hinted at inductive arguments for the existence of God; that is, I would expect them to accuse themselves of the very things they accuse me of — appeals to ignorance, personal incredulity, and the like — before accepting the POE as evidence against God. For every atheist that truly applies this skepticism to his own argument, I take no exception to their rejection of God.
Moreover, inductive arguments often fall prey to emotionalism, and this fact is exacerbated with subjects such as evil. Very often an atheist's rejection of God is based on emotionalism combined with the problem of evil. I think this is self-evident with regard to your greater question as to why some people disbelieve, and I would guess that it is a common path trodden by those deconverting from theism to atheism. Again, if any of your readers have taken the intellectual steps to ensure this is not the case with their thought process, and still remain convinced, I take no exception. In general, I take no exception to honest, well-thought through belief or unbelief.
So, properly framed, let's see where the discussion leads. The POE, the axiological argument, or perhaps "And Now for Something Completely Different."