Cares about

Jak się mówi? Julia doesn't care what happens to him czy Julia doesn't care about what happens to him?

bez about

Hmm...
W pytaniu sie daje about np, "Doesn't Julia care about what happens to him?", to dlaczego nie w twierdzeniu?

I don't care about what you think?

In informal situations we omit 'about' but proper grammar calls for it, I think. (-;

obie wersje sa poprawne

If a noun follows, about is needed:
I do/don't care about/Don't you/do you care about : (preposition required before noun: I don't care about those things.)

If a clause follows, no about:
I don't care/Do you care :
I don't care/Do you care who said that, what he thinks, why he is here, whether he is right or wrong, how far off the mark he is.

From Cambridge Dictionary:
Don’t you "care about" what happens to the children?

myślę, że jest tak, jak Aaric napisał.

podobnie ma sie sytuacja z innymi verb + preposition, a jednym z bardziej popularnych moze byc 'depend on'

To paraphrase your example, you could replace the "what happens to the children" with a noun phrase (e,g, Don't you care about the children/the children's future/well-being, etc.). There must be an overt or underlying object (usually a noun phrase) following the preposition.

How about "I don't care what happens to the children?" Now "What happens to the children" becomes a complement to the verb "care" and… the meaning changes.
"I don't care about…" entails lack of consideration or importance, whereas "I don't care" entails immateriality of choice or no preference or interest based on liking or partiality for something (for what happens to the children, for example).

Compare:
I don't care what Biden will do. (I don't care what he will or won't do. He belongs behind bars.)
I don't care about what Biden will do. (???)

So, it's not the informal style that lets you drop the preposition. It's not one of those "both are OK" situations either.