b: To Walk with 'a limp' means that you have something wrong with one of your legs or feet. In medical parlance we talk about 'short stance phase' which means when walking, one foot contacts the ground for longer than the other foot. This is usually because one foot , hip or knee is painful.
c: 'lore' means stories, history, legends as in 'folklore' - stories passed down (by mouth usually) from generation to generation.
d: To 'beckon' is to entice; but this word is, again, only used in a literary context usually. The meaning here is that the public schools appeared attractive to the narrator.