In New York too. "Schoolyard bully" sounds to me like a journalistic phrase. At any rate, my feeling is that "playground" is at least as common in the U.S. (though to me "playground bully" sounds decidedly odd). To "skip" rope sounds as right to me as to "jump," though the rope itself is always a "jump-rope." I think "off limits" replaced "out of bounds" only after WWII.

"Back in the day" bugs me because for some reason every TV newsperson suddenly began using it at the same time for no good reason. I still prefer "back in the old days" or, as my grandparents used to say, "in olden days" (or "times").