Answer: Hi, I'm Michael Dallen. I'm not a rabbi, but I did write The Rainbow Covenant: Torah and the Seven Universal Laws, and I help direct the First Covenant Foundation (1stcovenant.org). So I'm the one at e-Kiruv who usually gets questions like yours. Not that I'm complaining. Quite the contrary. These are good questions. I'm just trying to introduce myself to you, and direct you to a place, and book, where you can go for further answers.
You may be interested in knowing, first off, that when it comes to belief and things like proper or improper theology versus action, or how one chooses to actually live his life, the Torah teaches that "All is according to one's deeds." Not that intention doesn't count at all and not that prayer and belief don't matter, because they do, but, in this life, how one actually lives and acts, even though that's largely dependent on thoughts, beliefs and feelings, is the more important thing by far.
We often say that we pray not in order for God to save us but to make ourselves worth saving.
As for me personally, when I meet a non-Jew who rejects the non-gods that are worshiped by most nations, I don't see an atheist but, I hope, someone with the independent wit and the sound intelligence to turn away from idols. Neither does it bother me, particularly, that there's no recognition or appreciation of HaShem, the God of Israel (Who is, not so incidentally, my God). That's because:
1) He has been so badly misrepresented so often, as - God forbid - as an absolutely horrible Being, cruel, capricious, and just generally vile, or, on the other hand, as a sort of pet god that lives in people's pockets, their "best buddy," that most people with any sense fear anyone who talks a lot about whatever they call god; and
2) Non-belief in false gods, or in a god who is supposed to be God but isn't, like a) the Mohammedans' conventional imagining of a Jew-hating Allah, or b) Christianity's conventional conception of a hard and unloving god the father (who is pleased to bring people into the world "utterly depraved," as Christian doctrine has it, bound to suffer "conscious torment for all eternity" unless they "accept Jesus" and are "born again"), is usually necessary before one can begin to appreciate HaShem, the God of Abraham and Israel.
Allow me to say, parenthetically, that I disbelieved in what I thought was God - the invisible Being at the core of Conservative Judaism, whom our Conservative Movement rabbis, as it turns out, mostly didn't believe in either (!) - until I began to appreciate the wisdom and, particularly, the liberating, slave-freeing, historically demonstrated depth and power of the Torah.
I was long grown up, had a deep voice and was shaving before I started to appreciate that the God of Abraham wasn't anything like the "God Within" (like indigestion?) or the personally unaware cosmic force ("Cosmic Muffin") or the mean and horrible old man ("Hairy Thunderer") that I thought He was supposed to be.
You're right that God and allegiance to God is part of the basis of the Noahide/Noachide Law, and much of the point of the Law, but they aren't the whole thing. Every one of the Seven Laws, and all the myriad details of the Law - like, for instance, giving charity (giving charity being the reverse of the prohibition against larceny) and acting to save a human life in danger (which is the reverse of the prohibition against murder) - makes perfect sense logically and intellectually; they would all be incumbent on every righteous person at all times even if, God forbid, He were absent or not Himself.
I just spoke above about how badly the character of HaShem has been misrepresented: while foreign religions (as well as anti-religious movements) have been discrediting and defaming Him, the horrible off-putting nonsense that's still commonly attributed to the Noahide Law, even by Torah scholars who REALLY ought to know better, is probably even worse. In fact, these ghastly misrepresentations of the Noahide Law help explain why so many people have such fearsome and wrong ideas about God.
Anyway, bottom line - time is precious and I don't want to waste your time, or mine, by over-responding to your question - you can be a wonderfully good and excellent human being and not believe in any god. It all depends on what you do. Thanks to Moses and the prophets and the Sages of Israel, we happen to know what God loves, and - therefore - what is good, and that is, generally [drum roll, please] to do justice and righteousness and walk modestly, sexually, intellectually and physically, all through the course of one's brief span on earth.
He hath shown ye, o man, what is good; and what does the Lord [HaShem] require of thee, but to do justly, and to love mercy, and to walk humbly with thy God? - Micah 6:8
Happy is he who comes to the Hereafter possesssed of learning. - Talmud