What are three examples of an "American Romantic hero" in James Fenimore Cooper's novel The Last of the Mohicans?

According to the criteria you helpfully provided, at least three
characters in James Fenimore Cooper’s novel The Last of the Mohicans can be
classified as “American Romantic heroes.”  Those three characters are the
following:

• Natty Bumpo (“Hawkeye”):
  Hawkeye is very familiar with nature in general and with the forest in particular.
  He knows how to survive in the woods and is also skilled in handling a canoe on the river. He and
  his two Native American friends, Chingachgook and the latter’s son, Uncas, actually escape from
  danger at one point by floating down a river, thus demonstrating their resourcefulness and
  bravery. Hawkeye and his friends are also heroic in the way they attack the Hurons who have
  captured some white soldiers and two white women. Hawkeye is skilled in tracking animals and
  people, as when he traces the location of the white women after finding a veil belonging to one
  woman hanging from a tree. Hawkeye shows heroic bravery when he wears a disguise and manages to
  sneak into an enemy camp. As the novel concludes, Hawkeye uses his rifle to slay Magua, a Huron
  who is the major antagonist of most of the heroic characters in the
  book.

• Chingachgook:
  a brave Mohican chief, Chingachgook is a loyal friend to Hawkeye and courageously
  helps Hawkeye whenever the latter needs assistance. Chingachgook is a determined fighter and, as
  is customary, scalps some of the people he kills. When his own son, Uncas, is killed at the end
  of the novel, Chingachgook finds comfort in Hawkeye’s
  friendship:

readability="13">

Chinachgook grasped the hand that, in the warmth of
feeling, the scout had stretched across the fresh earth, and in an attitude of friendship these
two study and intrepid woodsmen bowed their heads together, while scalding tears fell to their
feet, watering the grave of Uncas like drops of falling
rain.

• Uncas:
  Uncas, a Mohican brave, is young, good-looking, courageous, and romantically
  attracted to one of the one of the main female characters, Cora. He uses his own familiarity with
  nature to help find her after she is kidnapped. He is a good and loyal friend to Hawkeye and a
  worthy son of his father, Chingachgook. Uncas assists his father and Hawkeye in all their heroic
  schemes, and when he is captured by their enemies he manages to use a disguise to help free
  himself from captivity. As the novel concludes, Uncas dies in hand-to-hand combat while trying to
  rescue Cora.

Hawkeye, Chingachgook, and Uncas thus
display many of the traits you listed as characteristic of an American Romantic hero: Uncas is
youthful; all three men have a strong sense of honor based on principle; all three display
knowledge that comes from intuitive learning, experience, and common sense rather than from
formal schooling; all three love nature over life in towns; all three are brave; Uncas is
handsome; and Hawkeye in particular seems to possess almost superhuman abilities, as when he
manages to shoot, from a long distance, a Huron who was just about to stab a
friend.