In Superman III the bad guys know about it from reading an interview with Superman.
I used to read Superboy (not so much Superman) comics as a kid in the 1960’s. I remember him shooting those mini-me’s from his fingertips – IIRC, as least in the telling I saw, it was one of the unpredictable but transient Red Kryptonite effects. I don’t recall the alien spaceships. That must have been a different (Superman?) tale. They did tend to recycle their plot-lines between the two series.
As for the movies: When I saw Superman 1, I always assumed that his act of turning time back (even against Jor-El’s advice) would somehow break the universe and let Zod et al out of the Phantom Zone.
There was also that throw-away scene, where Luthor’s moll takes the kryptonite necklace off of Superman in the pool and tosses it over her shoulder, whereupon it falls down a storm drain. Why did they put that in? I assumed at the time that the script writers MUST have a plan for that to re-appear in some sequel.
I missed Superman II, IIRC. Did that necklace down the storm drain ever play any further part?
ETA: BTW, in all the Superboy (and occasional Superman) comics I read in the 1960’s, I don’t recall ever seeing villain named Zod. Was he a new character, first appearing in the Superman 1 movie?
The necklace went down the sewer, I assume, to illustrate it being a safe distance from Superman.
Zod first appeared in the comics in 1961.
However, the other two evil Kryptonians were created for the movie, and have appeared in the comics since then.
I assumed that the necklace went into an old lead pipe, and the lead blocked the kryptonite from harming Superman.