For those who are interested, the “rhyming things with words beginning with ‘schm-‘” comes from Yiddish; specifically, a play in the Yiddish theatre that enjoyed huge popularity in the late 1870s (שמענדריק, אדער די קאמישע חתינה) wherein the title character, a bumbling fool, is called Shmendrik. The moniker Shmendrik is a play on the actual name Mendrik, which is itself a diminutive of the name Menachem.
As with many other aspects of Yiddish culture, the “__, shm__” formation caught on and has been adopted by the non-Jewish English speaking cultures of the world.
This is as cool as the first post!! I didn’t know the part about the play.