"I have discovered the God equation. God is not a kick. It is a way of life"
If you want to understand man's relationship to God, I think there is no better metaphor than that of addiction. Though I think most of life is best understood as the machinations of addiction. Before I start thinking, and reasoning, and looking, and talking, there is an unsated lust for some sort of reference to God. It is twisted, dishonest, irrational, absurd, impossible, individual and indespensible - just as it is for those who don't believe. You will lie to others, and you lie to yourself. And once you believe in God, you are no further forward. You have no better idea how to live. Eternity is not necessarily appealing. God is not necessarily good, and belief in God is not necessarily good for you. But too much of yourself is bound up with it; it is as fundamental as the desire for air, and more life. As rationally crappy as life is, there aren't many people who have killed themselves as a point of reason. The addiction to air comes first, and reason follows along after it.
Atheism is precisely the same. God is never a conclusion - whether you believe or disbelieve. The desire for something rational, tangible, is likewise best characterised as addiction - reason is pathological - evidence, as a notion, is a symptom of disbelief, not a cause. The disbelief started prior to analysis - it looked at the world and thought 'things couldn't be otherwise, there is no God, look at it'. The view of life that says 'Life is short, man is capable of much, after death is nothing, there are no absolutes' is intuition; the workings of something like the aesthetic sense. Reason can't get you to that view, but it can bolster it, verify it - you can invoke it, so you can pass the responsibility for your intuition onto something outside yourself. To give it some sensible, justified, form. There is nothing less attractive than an atheist saying 'I would believe if only there were evidence'. The lust for rational evidence is like Christians invoke the authority of scripture to get responsibility for their intuitions outside themselves, and to give them a coherent, authoritative form.
As for experience of God, it's certainly never something that couldn't be explained another way, if you had the will for it. For me, belief in God is roughly equivilant to love of life - that it has meaning. It's probably also belief in the magical, anti-natural, power of goodness. Most people become infinitely entrenched in their hatred of life; it's usually introverted, and subtle. People become, by degrees, autonomous, selfish, uncharitable. As, as I've had too much experience of lately, life shows its harsher self - people close to you die untimely, random, pointless, meaningless deaths - these things become easier, more justified. By the time you meet a fully developed human they tend to have outgrown playfulness, worn out by Time. Experience of God is in the tiny things - the small acts of goodness have a seemingly impossible softening power. Man is the creator of meaning. Despite its relative tinyness, to the harshness of life, goodness and meaning spread and multiply far beyond where they seemingly ought to go. In looking at its workings, its small sublimity - small, but infinitely unlikely changes in people, say - you get an impression of the sublime, if you have the will for it, not unlike the appreciation of valleys and chopin. It is is a completely natural response, but belief in God is not super-natural.
I don't think there's an extra set of feelings that could be called the God-feelings. Once addicted to God, all is God. Just like when addicted to nature, all is nature. It is not that God is superimposed on life; you just live out your life normally - it's just that life is that thing created by God. I'd love to stop believing in God. It's frankly tedious. But God exists, it couldn't be otherwise. Good has a redemptive power, life has meaning, we are creators, we are responsible for each other - these things are belief in God, I think. I couldn't get to atheism other than by conversion - something that involves far more than rational analysis. When someone has been a liberal for years, and they become a conservative (or vice versa), it is not rational analysis that has got them there. Their fundamental intuitions about The Way Things Are has changed - they have created them anew - and their reasons, and evidence, and arguments, follow along obediently. Even if God wrote 'I exist' in stars, it wouldn't help. My favourite atheist, other than Nietzsche and Camp, is a guy in my work named Boab. Back in my Christian-apologetic days I tried to talk to him about evidence for the existence of God. He said 'Mate, I don't give a flying f*ck about evidence - there is just no God, it's all a load of sh*te'. That's the most honest and responsible atheism that I love the most. Evidence of his existence couldn't convince you He was worth trusting, as the rest of how life works wouldn't change - it certainly couldn't convince you that you should re-arrange your life to glorify his, or any of that. If you are a decent, conscious, thinking atheist, just like if you are a decent, conscious, thinking theist, belief in the existence of God is utterly superfluous - it just is what it is and couldn't be otherwise. The harder questions - the ones you bind yourself up with answering - are the ones that follow.
God is irrefutable, because all the worthwhile references to God (or the worthwhile complete lack of references to God) happen way before and way after the games where the word 'refuted' has any weight take place.