IMPORTANT: I have provided a translation for each sentence, but DO NOT be reading my translations as you read my explanations. Instead, you should be looking at the original sentences in Japanese. Otherwise, some of the explanations might not make sense as it is not my translations that I am trying to explain.
わたしは[旅行]{りょこう} に [行]{い}きません。 = "I am not going on a trip."
"Direction?"
No, it is expressing "purpose", not direction. 旅行 is not a specific place or direction. Had the sentence been 「わたしはNYCに行きません。」, then the usage of the に would have expressed "direction".
8[時]{じ}の[電車]{でんしゃ} に [遅]{おく}れてしまいました。 = "I was late for the 8-o'clock train."
Object/target of action.
バスの[時間]{じかん} に [間]{ま}に[合]{あ}いませんでした。 = "I was unable to make the departure time of the bus."
Time of action.
電車 に [傘]{かさ}を[忘]{わす}れてしまいました。 = "I forgot my umbrella in the train."
Place of action.
"Why not use で instead?"
We just do not; You would sound pretty strange if you used で in that context. "Forgetting a belonging in the train" is not an "active" enough action for us to use で. It is more like the lack of an action of bringing the umbrella with you when you get off the train. If, however, you ate, danced, took pictures, used the phone, smacked a friend, etc. in the train, you would need to use で.
あの[先生]{せんせい}は[学生]{がくせい} に [人気]{にんき}があります。 = "That teacher is popular among the students."
Object of comparison or assessment.
画稿の近くです、子供 に 注意ください。← This makes no sense. Do you even know what 画稿 is? We do not say 注意ください, either. It should be either 注意してください or ご注意ください in natural Japanese.
I have taken the liberty of thinking the sentence was meant to be something like:
「[学校]{がっこう}の[近]{ちか}くです、[子供]{こども}に[注意]{ちゅうい}してください。」 = "School nearby. Drive carefully."
Object of action or attention.