Short answers, pulled from the story.
Walt Whitman first saw Abraham Lincoln on the 19th of February 1861 when the president-elect traveled through New York City. The poet estimated he saw Lincoln about twenty to thirty times between 1861 and 1865.
Walt Whitman never met Abraham Lincoln personally despite seeing him many times at White House receptions. They shared similar views on slavery and the Union but remained strangers throughout the Civil War era.
Walt Whitman wrote Hush'd Be the Camps To-Day, O Captain! My Captain!, When Lilacs Last in the Dooryard Bloom'd, and This Dust Was Once the Man following the death of Abraham Lincoln. He published these elegies in collections such as Sequel to Drum-Taps and Leaves of Grass starting from 1865.
Walt Whitman gave a series of Lincoln Lectures from 1879 to 1890 that centered on the assassination and covered years leading up to and during the American Civil War. These events included poetry readings and discussions with friends like Bram Stoker.