catlinさん、こんにちは!
I will give explanations for each below.
1.ジョンさんは日本語の宿題をしないで辞書を使(い)ます。
Mr. John can use Japanese dictionary without doing Japanese homework.
2.新しい漢字を覚えないで、漢字を書きました。
One wrote Chinese characters without remembering new Chinese characters.
3.ジョンさんは昨日よる、寝ないではをみがきま(す⇒した)。
Mr. John brushed his tooth without sleeping last night.
4。.ジョンさんは自転車をなおさないでたすけます。
Mr. John helps without reparing the bicycle.
Seeing above, 1,2 and 4 seem to have contradiction in meaning. For example, as for 1, Mr. John will not use Japanese dictionary if he don't do Japanese homework. I think 3 has no such contradiction.
But frankly I don't think this type of question to be good. For all of them can't be wrong in strict meaning. As for 1, it can happen that Mr. John use the Japanese dictionary without doing homework, like writing Japanese letter to friend.
If possible, I want to know the explanation given by the teacher when you got..:P