As to the first part, I seem to recall reading that bare-knuckled boxing emerged in the 19th century as a way for gentlemen to settle disputes without the danger of a duel with sword or pistol. Is this right? And as the second, I get the impression that it remained culturally acceptable in the west for two men to engage in a "fair fight" until fairly recently: say, the last thirty or forty years. When did this begin to change and why?
Boxing does become incredibly popular as a 'gentleman's sport' from the late 18th century, reaching its fashionable peak between c.1810-30. That said, it was still largely regarded as a spectator sport, with working class men doing most of the actual fighting. Pierce Egan, an Irish sportswriter living in London, wrote a popular series of articles which were printed in several volumes between 1813 and 1829 as Boxiana: or, the science of pugilism ancient and modern, which consolidated the status of boxing as a fashionable sport. You can see a representation of an 1820s boxing match here, taken from Egan's other major publication, Life in London.
That said, I haven't come across any accounts of elite men settling their personal disputes by way of a boxing bout. It may well have been done among boys - William Thackeray, writing in the 1840s but looking back to the 1800s-1810s, put a fistfight between gentleman schoolboys (Dobbin and George) into the plot of his novel Vanity Fair.