Ayn Rand -Faith vs Reason

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Partially correct. While both Ayn Rand and laissez-faire capitalism call for complete separation of economics and government, Ayn Rand atleast acknowledges that it is impossible in the current scenario. The reason for that would be:
1. Certain services that the government currently provides would not be so easily privatized, such as roads, city infrastructure etc.
2. A source of revenue for maintaining the government would be required - which would automatically put it back in economics.
3. There can be practices is economics that can be coercive and therefore call for government involvement. This is a point that I think Ayn Rand missed.

#3 is putting it lightly. The primary driving force behind laissez-faire unregulated capitalism is profit and capital. The ultimate guiding forces are not principles per se but sheer market forces. It is a kind of "natural selection" in that companies which are profitable survive and those that aren't do not. For instance if it is not profitable for a company to clean up an oil spill they will simply not do it. Monopolies would form. Nevermind workers rights and things like setting a minimum wage or collective bargaining. Regulated capitalism works, unregulated capitalism would be a nightmare.

These are the reasons why pure Laissez-Faire seems impossible, atleast currently. However, it has been adopted as an ideal in many scenarios, its principles have been applied and a great deal of progress has been observed in those cases. Its no miracle that the countries with lesser government involvement in economics boast of better quality of life. While pure Laissez-Faire has never existed, the concepts behind it (such as free-market) have been put into action and observed to work.

Can you name a couple example countries? To my knowledge the countries with the highest standards of living have significant socialized government programs such as education and healthcare.

Your fear of stratification of wealth and worse living conditions for the poor springs from the incorrect assumption that the total wealth would remain constant. If the total wealth were to remain constant, any increase in wealth of one section of the society would necessarily come at the cost of another section. However, if the total wealth increases, then all the sections can get wealthier even if the wealth gap between them increases.

Even under our current regulated US economy (though significantly less regulated than the prosperity of the 50s) we have seen a stagnation in wages for low and mid level employees and an exponential growth in the wages of higher level management and CEOs. Not to mention growth in their salaries during a huge recession. The 1% continued to amass wealth even when the vast majority of the remainder of the population saw a stagnation or reduction in their wealth, employment, and wages. This is in a regulated economy. In an unregulated system there would be no checks to ensure that the rich don't simply completely screw over the poor once and for all. Even if total wealth increases why would those who set the wages and are responsible for the distribution of the wealth not simply continue to amass capital and profit for themselves?