I'm actually a law student, not a historian, so I hope I'm allowed to post here. I cannot speak to the religion and morality portion, but know quite a bit about the founders/American dream portion having just finished my Constitutional Law course.

This is a subject that is closely intertwined with competing views of what the Constitution means, slavery, the Civil War, and the Reconstruction Amendments. The debate over slavery was a hotly contested subject from our nation's very beginning: the Constitutional Convention. Ultimately it was not constitutionally outlawed, as a concession to the southern states, but opponents of slavery continued their efforts to outlaw it. Attempts by Congress to outlaw slavery came to a head with the Missouri Compromise, and the infamous case of Dred Scott v. Sandford. This opinion was hundreds of pages long, but rested on the idea that slaves were property, and attempts to outlaw slavery was a deprivation of the slave owners property. This idea, called "substantive due process", is hotly contested by law professors and academics even to this day. Either way, Capitalism and freedom of property became the rallying point of the southern states before and through the Civil War.

Fast forward some years, and we arrive at Lochner v. New York. This case resulted from attempts by the New York legislature to cap the number of hours bakers could work. To grossly simplify Lochner, The Supreme Court invalidated the New York law using similar reasoning as in Dredd Scott: attempts to curtail the number of hours that bakers could put in was a deprivation of their liberty (specifically, their liberty to earn a living). This kicked off the infamous "Lochner era", where The Court used the new 14th Amendment's due process clause ("nor shall any State deprive any person of life, liberty, or property, without due process of law") to strike down all sorts of laws.

The Lochner-era judges had all gone to law school during the Civil War era. This broad idea of Constitutional liberty was something that had been developed in response to the anti-slavery movement, and was first used by The Court in Dredd Scott. During the time period from Lochner to the New Deal, The Court enforced this broad notion of liberty, and among these was the idea of unrestricted Capitalism. In Justice Holmes' dissent to Lochner v. New York, he lamented that The Court was writing a laissez-faire idea of Capitalism into the Constitution through judicial activism.

Lochner was overruled during the New Deal era, and of course Dredd Scott was overruled and slavery outlawed by the Reconstruction Amendments. However, their basic notions has persisted to this day: that the US Constitution espouses a basic notion of liberty that neither the federal nor state governments can infringe upon. While economic freedoms and concepts of Capitalism are no longer enforced by The Courts as a Constitutional Doctrine, they have remained ingrained in our culture. And these ideas of "fundamental rights" have resurfaced recently, specifically in the areas of individual privacy (Roe v. Wade, and Lawrence v. Texas).

Others are more educated in this field, and can better speak to when it became intertwined with religion and morality. But the idea that "America is a Capitalist nation" can be traced back to the debates before the Civil War, as a response by the southern states to attempts by the northern states to outlaw slavery through legislation, instead of through a Constitutional Amendment.

This paper is a good historic overview of the Lochner era, and the thinking at the time of the Courts and country: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=395620

I've been exploring this and some related questions, and a tool I'm finding hugely valuable is the Google Books ngram viewer, which can be used to find when specific words entered into widespread usage within published documents from 1500 to the near present (the record ends at 2008 for now).

Looking at the prevalence of "capitalism" from 1800 to 2008 we find that there's some growth in use of the term through the 1920s, but it truly emerges following the 1929 stock market crash. Often in the form of criticism (e.g., George Bernard Shaw's The Intelligent Woman's Guide to Socialism and Capitalism.

I find it interesting that a number of words and phrases which are thought to be enduring and eternal American values largely emerge only in the 20th century. A few of interest being democracy, capitalism, free market, laissez faire, technological progress.

Note that democracy in particular emerged during WWI, it was actually part of the campaign to sell war bonds, and specifically, of "making the world safe for democracy".

And that tied in with the start of "public relations" which replaced the earlier concept of propaganda and Edward Bernays.