Yes

It isn't a common expression. It doesn't have any standard or precise meaning. It might help if you gave us the context in which you saw it. A Google search turns up an essay in "The New York Times" by David Brooks which opens "During the middle third of the 20th century, Americans had impressive faith in their own institutions." Is that where you saw it? President Franklin D. Roosevelt began a series of government initiatives called the "New Deal" in 1933, which built trust in the U.S. government to help the common people during times of economic difficulty. The United States involvement in the Vietnam War escalated in 1964 with the Gulf of Tonkin incident, and after about 1964 the Vietnam War destroyed a good deal of trust in the US government. It might be reasonable to think of 1933-1964 as an era of faith in government, and that would be roughly the middle third of the century. Brooks might have something like that in mind.

Yes. Or perhaps the 1940s and 1950s. It's just an approximate time-span.