From what I could google, the barrel organ was sort of like a player piano. The tunes played on it were determined by a barrel shaped part which held the musical notes to be played. All the "organ grinder" had to do was turn a crank. I belive the tunes could be changed by switching barrels. I have the impression that Chesterton wanted to convey a sense of the common man with this rather mundane musical instrument. There is something anti-eliteist about an instrument played by an unskilled laborer rather than a talented musician.
Next comes the quote from the Song of Roland. Chesterton mentions the Old French saying : "Pagens ont tort et Chretiens ont droit", which translates : Pagans are wrong and Christians are right. In my humble opinion the original French has more of a ring to it than the simple English translation. Syme, like Roland is taking a stand against what he sees as pagen influence. This does not bode well for Syme since Roland was killed by Saladin. Split in two, as a matter of fact.
Chesterton has us sitting on the edge of our seats as Sunday glares at him during the whole breakfast. When Sunday declares the presence of a traitor we assume the worst. After all, this chapter is entitled "The Exposure". When Gogol turns out to be Sunday's person of interest, we breath a sigh of relief along with him.