If(?) the Order of the Green Hand predates the arrival of the Andals and their knighthood, do you think that Barrow knights could have similarly evolved from a group sworn to the First Kings of Barrowton, only later becoming “knights”? – RSAFan

There’s a couple of possibilities. The first, and most likely, is that the Order of the Green Hand post-dated the Andal invasion, since the Gardeners from whose sigil it drew its name managed to thrive during that invasion. The second is that the Order was previously a group of “sworn swords” similar to Northern cavalry, and adopted the trappings of chivalry after the Andals. 

As for the barrow knights, there are also a couple possibilities. The first is as you suggest. The second is that the term barrow knight is used similar to “hedge knight,” indicating an independent mounted warrior of the barrowlands who made their holdfasts in the artificial hills. Indeed, given the real-world use of the term to describe prospectors and archaelogists who dig in prehistorical barrows, it might be a derogatory term similar to “hedge knight,” indicating a warrior whose wealth and position comes from robbing the tombs of the dead. The third is that GRRM likes to play as the Vampire Counts from Warhammer Fantasy.