Well, I'm not a native, as you can see, and my results come from Google. I haven't found it registered under "idiom" in Cambridge.
First it's literal: to throw a stick in the direction of someone.
Then you have "to throw a stick into something", it's to try & prevent the success of something.
"More than you can throw a stick at" seems to mean "more than expected"
And then you have the "throw(ing) stick", it's a boomerang, the curved piece of wood that's supposed to come back to you after you've thrown it.
Anyway, the normal order of complements is "to throw what/at - to whom/what"
That's all I can do for you, Anon, sorry, others will surely give their opinions later...