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About context. It is a sentence from an anime series (charlotte). One of the characters proposes their actions regarding other character (Ayumi, 歩未ちゃん).
Here we have 歩未ちゃんへの注意 (attention to Ayumi). And the sentence says to stress (...を促す) this attention. So I guess the meaning is to make the teacher watch Ayumi closely but how does に work here? Is the idea similar to something like "ジョンに花をあげました。" But instead of flowers we have attention and instead of giving (あげる) we have stressing (促す)?
Basically yes, 教師 is the indirect object and 歩未ちゃんへの注意 is the direct object of 促す: urge teachers to watch Ayumi carefully.
Just like this case, if the direct object is a noun meaning some action, it often corresponds to English verb + obj. + to inf.
程度 here is like "just...". E.g.,
運動しますか Do you do any exercise?
散歩する程度ですね Just walking sometimes.
