How are such things as liberty, egalitarianism, individualism, populism, and laissez-faire individually connected to the Western time period?
egalitarianism: belief in equality of all people
individualism: all focus is on the individual
populism: belief that the government should make every decision based on the concerns of the majority of the people
laissez-faire: belief that the government should intervene very little in economic affairs
western era: roughly 1860s-1890s
Looking at how you have defined the "western time period," I assume that you are asking about this specifically in the context of the United States. If so, all of these things are connected to the western time period through (as Frederick Jackson Turner argued) the influence of the frontier.
During the time period you are asking about, the United States was characterized, as much as anything, by westward expansion. This was the time of cowboys, the Indian wars, and the "Old West" in general. Turner argued that this experience of going out into the frontier shaped the American character. He argued that it generally created an American ethos in which people expected to be self-reliant. All of the things that you mention flow from this basic attitude. All of them come from the idea that individual Americans were able, on their own, to overcome the hardships and demands of creating and living on frontiers.