I do not think that any did. A person born in 1845 would have been 70 in 1915. Many characters in the Old West lived until well into the 20th Century, but if any served in WWI as a soldier they were most likely Europeans who had been in the Union Army. I do not know if any were so documented.

Not that I am aware of. You can find veterans in the Spanish-American War, but WWI is over a decade beyond that. The best possibility that came to mind is Johnny Clem, a drummer boy who was born in 1851 and served with great distinction during the Civil War. He returned to the Army as an adult but retired in 1915, so two years short of WWI.