I'm curious if these figures retained actual personal bodyguards (as opposed to soldiers who happened to protect their residences), or if it was simply hard enough to access them that it wasn't considered necessary. Given that much of the male nobility consisted of trained fighters, many of whom (one presumes) were extremely loyal to the monarch, perhaps they felt safe enough with just them hanging around? It struck me that in all my various readings about various English monarchs, I'd seen mention of the regular proximity of friends, office-seekers, serving staff, and even jesters, but never of people whose job was simply protection.
You're correct in assuming that many of the household attendants of the king would have martial experience. When the royal household went to war, many of its members (including professionals such as armorers and tailors) also fought, often as archers. So a fair number of the various hangers-on you mention would have combat experience to some degree.
However, English kings from about 1300 on also maintained men who were professional bodyguards. Prior to Richard II, it seems that the usual number of bodyguards was 24 mounted archers. In 1360, the 24 men of the king's bodyguard at Calais were each issued 1 "painted bow," a hundred arrows, and ten bowstrings. Edward III's guard were apparently issued with some kind of special quiver, which is unusual because English archers typically carried their arrows tied to their belts. Richard II expanded the institution of "archers of the crown." He not only increased the actual number of archers of the crown, but created a separate bodyguard of at least three hundred men drawn from Chester county, which Richard directly ruled. It seems that the massive expansion of the Cheshire archers was done in an effort to skirt Parliamentary regulations, which attempted to limit the king's bodyguard of archers to the traditional 24 men.
There is a certain amount of doubt as to the nature of their service. Some of these men had other duties (with titles like "yeoman of the chamber" or "groom of the chamber"), but that does not exclude them from the capacity to commit violence in the name of the king. Their socioeconomic background is also unclear. Since their wages often went unpaid, it is presumed that many, if not all, of these men would have had some other means of supporting themselves on a day-to-day basis. Richard II's Cheshire guards, being much larger in size, had a very wide range in status, ranging from men with substantial property to men who owned no land but a handful of livestock.
Sources: The Soldier in Late Medieval England, by Adrian Bell, Anne Curry, Andy King, and David Simpkin
Investigating the Socio-Economic Origins of English Archers in the Second Half of the 14th Century, Gary Baker, Journal of Medieval Military History, Vol. XII
The Medieval Inventories of the Tower Armories 1320-1410, Roland Thomas Richardson, University of York
The Byzantine Emperors had the Varangian Guard, recruited from Scandinavians and Anglo Saxons, from its founding in 988 by Emperor Basil II, until the mid 14th century.
The French Monarchs had a body guard from 1440, when the Scottish Guard was formed.
In England, Henry VII created the Yeomen of the Guard in 1485, after the Battle of Bosworth. The Yeomen at one time kept 40 guards on duty in close proximity to the King during the day, and 20 at night. They still exist, but have only a ceremonial role these days. One of their duties being to inspect the cellars of the Palace of Westminster before the State Opening of Parliament every year to ensure that no one has tried to emulate Guy Fawkes, and stack the cellars with gunpowder to blow up the monarch and the Parliament.