What deep-rooted american ideals and beliefs are found in the mythic status of western cowboys?

Those probably were freedom, independence and self-reliance. The cowboys represent brave and unflinching individuals who conquer unknown and often very dangerous new frontiers of the Wild West. They brave all the dangers with nothing but their wits and their skill with a firearm which usually is a pistol. They are free individuals in a free open space who are guided by their instincts and their morals.