Lines 1-2: Falconing was an activity that is associated with medieval times. People with enough wealth – that is, feudal landowners – often built aviaries where they kept birds to use for hunting. The most common were falcons and hawks. As such, this activity is associated with violence, but not the kind of uncontrolled, chaotic violence that characterized World War I. Falconing was a noble activity in which the bird was tightly controlled by its master. Obviously, that is not the case here. The reference to falconing should also be read as a symbol of the virtues of the Middle Ages: order, tradition, strong religious faith, unified government, and "civilized" warfare.
Line 19: The word "stony" has many connotations here, but one of them might refer to the Middle Ages again. After all, when you think of a medieval structure like a Gothic cathedral or a castle, you probably think of a big, strong, stone building. "Stony" here suggests something that endures and lasts a long time. However, we also call a person "stone-like" when they are incapable of feeling or reacting. In addition, Yeats is using the image of an infant’s "sleep" as a metaphor for the roughly 2,000 years between the First and Second Comings. It’s easy to forget that the Middle Ages lasted around 800-1000 years, from around the 5th to the 15th centuries in Europe. Therefore, it was the longest period of the last "twenty centuries," and Yeats might be using it as representative of the period as a whole.