The joke is a mild and uncomplicated one. The idea is that you pull the string and the doll kvetches (complains) about it. "Again with the x" is just an idiomatically Yiddish way of complaining.
Nothing to do with "no strings attached".
EDIT:
The whole point of a talking doll is that when you pull the string, it says something appropriate to the particular doll. Woody, the cowboy doll from Toy Story, would say things like "Somebody poisoned the water-hole!" or "There's a snake in ma boot!" Mattel Corporation got in hot water for manufacturing a teenaged-girl doll that would say, "Math is hard!"
So, a old-Jewish-lady doll would be expected to say, "Have a piece of fruit" or "Oy, do my kishkes [insides] hurt!" but all humor depends on partial reversal of expectations. If the doll merely complained, that wouldn't be funny because that's what you'd expect; if the old-Jewish-lady doll said "There's a snake in ma boot!", that wouldn't be funny because that just wouldn't make any sense.
To be funny, a joke must both confirm and confound the expectations: the old-Jewish-lady doll complains about your pulling its string, in perfect Yiddish idiom.
Comedy gold, Jerry!
Another powerful aspect of a joke: hyperbole. The doll should really say, "Again with the string? Oy, gevalt. Enough with the string already!"
Yoichi Oishi, no offense meant but I imagine you with a pronounced Japanese accent. If you tell this joke (which is not particularly well-known), aloud and at all well, to any Jewish American, that person will be telling his friends about you at parties for the rest of his life.
About 1985, I saw a Japanese stand-up comic on TV say (in an almost unintelligible accent), "Not many people know this but Japan have velly small army. Velly velly small. That why monsters always attack Tokyo first."
Still cracks me up.
It's a reference to the prophecy that King Arthur will return. The idea is that he was once king, and will be again.
As far as I know, T.H. White did in fact coin the English version of the phrase for his Arthurian book The Once and Future King, but you'll occasionally hear it adapted for other uses ("ladies and gentleman, the once and future champion!"), presumably as an allusion to the book. The original source is Sir Thomas Malory's Le Morte d'Arthur (the most enduringly popular rendition of the Arthurian myth), where the equivalent Latin phrase rex quondam rexque futurus is described as engraved on Arthur's tombstone.
Best Answer
"What would you with the king?" is an archaic construct (but of course common in Marlowe's time), meaning "what do you want with the king?", or "what is your reason for wanting to talk to the king?"