Final answer:
Theodore Roosevelt became president after President McKinley's assassination, introduced the "Square Deal" domestic policies focused on consumer protection, corporate regulation, and conservation, and was a strong advocate against monopolies, earning him the Nobel Peace Prize.
Step-by-step explanation:
Theodore Roosevelt, often referred to as "Teddy" or TR, was a dynamic figure in American history and a representative of the Progressive Era's spirit. The statements that correctly describe parts of his life and presidency include that he became President following the assassination of President McKinley, his domestic programs were dubbed the "Square Deal," and he was known for his conservation efforts regarding American wilderness and forests. He also won the Nobel Peace Prize for his role in ending the Russo-Japanese War and ran for President as a third party candidate in 1912 with his Bull Moose Party, which ultimately split the Republican vote. Contrary to one of the statements, Roosevelt strongly supported the regulation of big companies and monopolies (trusts), a position that was integral to his reputation as a "trust buster."
Roosevelt's presidency marked a shift in the executive branch, elevating the office's stature with his vigorous approach to leadership and policy. His commitment to the "strenuous life" reflected in his adventures and military service, alongside his progressive policies supporting consumer protections, corporate control, and conservation, established him as one of the first "Progressive Presidents."
While Roosevelt did not serve more than two terms as President and did not believe in selling American wilderness to oil companies, his influence and progressive policies set a precedent that would influence future administrations. His challenge to President Taft in 1912, though unsuccessful in securing the presidency, showcased his continued commitment to progressive ideals and his broadening political orientation towards the left.