Excellent Nage!!
Why did they wear red coats? Not having been in the Army myself, my first thought would be it was a socially acceptable way of cross-dressing from the seventeenth to nineteenth century.
Or; forget about the blood thing. Its a myth. By around the end of the seventeenth century most armies had given up their pikes and halberds and were equipped with muskets and supported by cannon and cavalry. Any battlefield very quickly became obscured by smoke. It made it much easier to tell friend from foe if everyone was colour coded.
Since a musket's effective range is around fifty metres there was no question of camouflaging anyone. This only started to matter around 1850 when the first rifles came into use with effective ranges up to three and four hundred metres.
Red had been worn by various British troops from around the mid-sixteenth century but came to be the standard colour of the uniforms issued to the New Model Army during the English civil war. Its an expensive thing kitting out an army and red dyes were cheap and the process simple. Britain was not the only nation to adopt red for uniforms either. However, the British army were one of the first armies to discard bright, colourful uniforms for combat after their experiences in India where Khaki became standard in the latter half of the nineteenth century. I think - but I'm not too sure - that the British army ( or part of it ) last fought in red coats in the Sudan in the late nineteenth century....er, possibly.
Red tunics and ladies underwear still play an important part in ceremonial uniform worn by the British Army to this day.
Edit; Jabo beat me to it. He just skimmed over the cross-dressing