You ask some interesting questions about Objectivism and Christianity: the differences and similarities between the two worldviews, Christian Libertarianism, Christian Anarchism, and Rand's politics, and my own view. I'll do my best, in a nutshell.
Rand is a committed atheist. She believes in absolutes, but not in a Giver of absolutes. Taking a page from Aristotle, she marks her own life as the highest possible occupation of her soul; not others, not the collective, and certainly not God. About every decision and every value, she asks "Will this promote, maintain, or enhance my life?" If so, it's a virtue. If not, it's a vice. After all, she reasons, without my own life I could have no values; my life is therefore the Supreme Value. A person must produce, or create value, in order to sustain their life. So, in all of Rand's fiction, the Producers are the Prime Movers, the heroes and heroines.
Mother Teresa, in Rand's view, was not a hero and neither was Jesus. In her view, both were moochers, and deserve scorn for their self-sacrifice. Ayn Rand was a passionate individual and a brilliant thinker, but I don't think she truly understood the Gospel. She railed on Christianity, but it was a shallow Platonic version that focused on Victorian religion and Kantian duty. Nothing like what John Piper calls "Christian Hedonism." To the extent that Rand advocated that individuals pursue their highest possible joy, I applaud and appreciate her. But her passion falls short; she "settles" like Lewis's "child making mudpies in the slums..."
Through a journey that began with cancer when I was 17, Jesus drew me to himself. The deepest longings I have felt throughout my life have only been met in Him. Still, my hunger for his presence grows. Today, I can't say that I'm an Objectivist, or even an Objectivist Christian. I would say that Objectivism finds its fulfillment in the Gospel because the truest and highest occupation of my soul is not myself, but God the Father, through Jesus Christ.
Objectivism and Christianity share a strong passion for life, a vibrant work ethic, a conviction around moral absolutes like honesty, integrity, and respect for the individual. Unlike many postmodern thinkers, both Objectivism and Christianity recognize objective truth and see truth as exclusive. The Objectivist becomes arrogant, because he has achieved something that has eluded his contemporaries. The Christian becomes confident based on the achievement of salvation, but humble, because he had nothing to do with it.
Rand's politics are economically conservative and socially liberal. Ayn Rand raged against the Libertarians of her day because they were anarchists; they believed that government had no place, and its authority should be avoided altogether. Today's Libertarians, like Rand, advocate for limited government. Rand believed that government's role is to protect the rights of the individual (the smallest minority in the world) from criminals at home and abroad. It is the function of our Constitution, she said, to protect the individual from government.
I suggest approaching today's new atheists (for whom Rand's thoughts are seminal) with three elements in mind. First, articulate their worldviews accurately and clearly, avoiding oversimplification and caricature. Next, identify areas of truth and validate, wherever possible, the values we hold in common. Finally, show how the Gospel values these truths even more highly than the atheistic worldview. While they may not be converted after the conversation, they go away wishing that Christianity were true.
I have attached an essay by John Piper entitled, "The Ethics of Ayn Rand: Appreciation and Critique." I think his view is accurate and balanced. There are two new biographies of Rand that hit the shelves in October. Jennifer Burns' "Goddess of the Market: Ayn Rand and the American Right" is focused on Rand's politics of capitalism; critical, but balanced. Even more critical, Anne Heller's "Ayn Rand and the World She Made" gets into the destruction she caused through her brilliance, eccentric passion, and emotional sterility.