Can't I say "무엇을 아이에게 가리칠 것이에요?"
You can. "무엇을 아이에게 가르칠 것이에요?" but I think it should be "아이들", not "아이" because of "the children"
If you were to do it without the next syllable I think it would be the child
아이한테 뭘 가르칠 거에요? Will this work too?
Why not "아이들을"?
i think it's because there's already 을 with 무엇을 so it would sound weird if you say 아이들을 무엇을
because what you are teaching is "something" not "children". you are teaching /something/ to them. so 을 is attached to the word being "modified" by the verb
The last sentence was wrong because I did not put 줄. Someone wrote that we must use 줄 if it directly benefits someone. So why isn't this sentence using 줄?