A couple of observations on this interesting thread...one of those that makes the Mudcat worthwhile--thanks for starting it, kat!

The rhyming words I learned to the song (Philadelphia, early 60s) are as follows: "He played knick-knack on my..."
1 - thumb; 2 - shoe; 3 - knee; 4 - door; 5 - hive (I never liked that one, but there it was); 6 - sticks; 7 - up in heaven; 8 - gate; 9 - spine; 10 - He played knick-knack once again.

Others probably have variations, though I suspect ours came from the 1958 movie, which I don't believe I ever saw. (I know I never saw the 1858 movie!)

Regarding "paddywhack": when I was a kid, when you did something particularly egregious (among other kids, that is), the penalty was to "get paddywhacks." This meant that the others would stand in two lines facing one another. The offender would run the gauntlet, while his "friends" whacked him on the butt as many times as they could. We never thought about the derivation of the term, that was just what paddywhacks were. But I wouldn't be surprised if it originally had something do to with whacking an Irishman, or at least pretending to. My guess is that the newer version of the song substituted that word for the original (padlock, or whatever) just because it sounded good.

Aloha,
Mark